Of course, having high rep or gold badge dupe-hammer on one Stack Exchange site should not automatically grant me the same or equivalent on another, but I feel it should grant some of the lower privileges - actual up/downvotes, close/reopen votes, ability to see deleted answers and questions rather than getting a 404, review queue abilities.
I'm aware that having more than 30k in gardening, for instance (I don't; it's just an example) does not automatically mean I'm any expert on Unix—but it does signify I'm likely to be reasonably adept at weeding out the truly bad questions, or separating the wheat from the chaff on new users/questions (see what I did there?;) and that I ought to be responsible enough to not spuriously or unthinkingly mark questions as 'needs detail' simply because I don't understand the topic.*

As an aside, this could be discussed as a separate issue if the idea itself gains traction in this Q&A. *I have quite a history of flagging those bot comments set by people who clearly didn't understand the question yet post that pointless boilerplate/rubber stamp anyway.
So, slightly straying from the point; that could be one new privilege at >30 - 50k anywhere (consolidated) - one-click removal of badly-applied bot review comments… would save a mod having to do them all.
This is the specific bot-comment I'm referring to above, which is so often badly-applied by someone who didn't understand the question, yet to someone who does understand it needs no further clarification. It's good to go as-is, if you understand the topic.
This is very often slapped onto new user questions too, where a little better individual guidance would help far more than this boilerplate/rubber stamp. I often find new users have responded to this comment in the mistaken belief that it's a real person, and they will get the message, so I have to spend time explaining what actually happened.

Or

Reviewer is really not paying attention.

I love irony - this just in

Of course, having high rep or gold badge dupe-hammer on one Stack Exchange site should not automatically grant me the same or equivalent on another, but I feel it should grant some of the lower privileges - actual up/downvotes, close/reopen votes, ability to see deleted answers and questions rather than getting a 404, review queue abilities.
I'm aware that having more than 30k in gardening, for instance (I don't; it's just an example) does not automatically mean I'm any expert on Unix—but it does signify I'm likely to be reasonably adept at weeding out the truly bad questions, or separating the wheat from the chaff on new users/questions (see what I did there?;) and that I ought to be responsible enough to not spuriously or unthinkingly mark questions as 'needs detail' simply because I don't understand the topic.

After comments basically asking "what's the point?" - this is one of my quietest groups. Most of the questions mean nothing to me because the subject matter is extremely broad, so my Q&A interaction is small. I am, however, perfect capable of spotting the completely off-topic or truly bad questions whether I know the answers or not. At this rate I'll be able to contribute to that in about another decade or so.
Flags add to the moderation effort; they do not reduce it, which would be the primary purpose of such additional privileges.

After comments basically asking "what's the point?" - this is one of my quietest groups. Most of the questions mean nothing to me because the subject matter is extremely broad, so my Q&A interaction is small. I am, however, perfectly capable of spotting the completely off-topic or truly bad questions whether I know the answers or not. At this rate I'll be able to contribute to that in about another decade or so.
Flags add to the moderation effort; they do not reduce it, which would be the primary purpose of such additional privileges.

Privilege consolidation

At the moment, the low bar to cross-platform 'trust' returns also low privileges. Perhaps this should be tiered. [This could also be combined with some of the other answers here, or taken to a new Question, as to precisely which privileges.]

As a user who has >30k on some stacks [& actually >300k network-wide], I find it somewhat restrictive if I join a new stack to be able to do very little in the way of 'routine' tasks I can do on the others.

Of course, having high rep or gold badge dupe-hammer on one stack should not automatically grant me the same or equivalent on another, but I feel it should grant some of the lower privileges - actual up/down votes, close/reopen votes, ability to see deleted answers & questions rather than getting a 404, review queue abilities.
I'm aware that having >30k in gardening, for instance [I don't, it's just an example] does not automatically mean I'm any expert on unix - but it does signify I'm likely to be reasonably adept at weeding out the truly bad questions, or separating the wheat from the chaff on new users/questions (see what I did there?;) & that I ought to be responsible enough to not spuriously or unthinkingly mark questions as 'needs detail' simply because I don't understand the topic.*

One practical consideration would be how this is applied. Currently, it would require the user to be handed a far greater number of 'trust' points when joining a new group.
I don't see how this would work; it's not particularly 'fair', it implies more tested knowledge & experience in the new topic than strictly necessary. It would be difficult to backdate.
An alternative might be to award it as a badge instead. 'Consolidated contributor' or something similar. I'm still not certain how simple this would be to backdate. It would allow tiering though, as there are already bronze, silver & gold tiers within the badge system.

After comments basically asking "what's the point?" - this is one of my quietest groups. Most of the questions mean nothing to me because the subject matter is extremely broad, so my QA interaction is small. I am, however, perfect capable of spotting the completely off-topic or truly bad questions whether I know the answers or not. At this rate I'll be able to contribute to that in about another decade or so.
Flags add to the moderation effort, they do not reduce it, which would be the primary purpose of such additional privileges.

As an aside, this could be discussed as a separate issue if the idea itself gains traction in this QA.
  *I have quite a history of flagging those bot comments set by people who clearly didn't understand the question yet post that pointless boilerplate/rubber-stamp anyway.
So, slightly straying from the point; that could be one new privilege at >30 - 50k anywhere [consolidated] - one-click removal of badly-applied bot review comments… would save a mod having to do them all.
This is the specific bot-comment I'm referring to above, which is so often badly-applied by someone who didn't understand the question, yet to someone who does understand it needs no further clarification. It's good to go as-is, if you understand the topic.
This is very often slapped onto new user questions too, where a little better individual guidance would help far more than this boilerplate/rubber-stamp. I often find new users have responded to this comment in the mistaken belief that it's a real person & they will get the message, so I have to spend time explaining what actually happened.

Reviewer really not paying attention.

Privilege consolidation

At the moment, the low bar to cross-platform 'trust' returns also low privileges. Perhaps this should be tiered. (This could also be combined with some of the other answers here, or taken to a new question, as to precisely which privileges.)

As a user who has more than 30k on some Stack Exchange sites (and actually more than 300k network-wide), I find it somewhat restrictive if I join a new Stack Exchange site to be able to do very little in the way of 'routine' tasks I can do on the others.

Of course, having high rep or gold badge dupe-hammer on one Stack Exchange site should not automatically grant me the same or equivalent on another, but I feel it should grant some of the lower privileges - actual up/downvotes, close/reopen votes, ability to see deleted answers and questions rather than getting a 404, review queue abilities. 
I'm aware that having more than 30k in gardening, for instance (I don't; it's just an example) does not automatically mean I'm any expert on Unix—but it does signify I'm likely to be reasonably adept at weeding out the truly bad questions, or separating the wheat from the chaff on new users/questions (see what I did there?;) and that I ought to be responsible enough to not spuriously or unthinkingly mark questions as 'needs detail' simply because I don't understand the topic.*

One practical consideration would be how this is applied. Currently, it would require the user to be handed a far greater number of 'trust' points when joining a new group. 
I don't see how this would work; it's not particularly 'fair'. It implies more tested knowledge and experience in the new topic than strictly necessary. It would be difficult to backdate. 
An alternative might be to award it as a badge instead. 'Consolidated contributor' or something similar. I'm still not certain how simple this would be to backdate. It would allow tiering though, as there are already bronze, silver, and gold tiers within the badge system.

After comments basically asking "what's the point?" - this is one of my quietest groups. Most of the questions mean nothing to me because the subject matter is extremely broad, so my Q&A interaction is small. I am, however, perfect capable of spotting the completely off-topic or truly bad questions whether I know the answers or not. At this rate I'll be able to contribute to that in about another decade or so. 
Flags add to the moderation effort; they do not reduce it, which would be the primary purpose of such additional privileges.

As an aside, this could be discussed as a separate issue if the idea itself gains traction in this Q&A. *I have quite a history of flagging those bot comments set by people who clearly didn't understand the question yet post that pointless boilerplate/rubber stamp anyway. 
So, slightly straying from the point; that could be one new privilege at >30 - 50k anywhere (consolidated) - one-click removal of badly-applied bot review comments… would save a mod having to do them all. 
This is the specific bot-comment I'm referring to above, which is so often badly-applied by someone who didn't understand the question, yet to someone who does understand it needs no further clarification. It's good to go as-is, if you understand the topic. 
This is very often slapped onto new user questions too, where a little better individual guidance would help far more than this boilerplate/rubber stamp. I often find new users have responded to this comment in the mistaken belief that it's a real person, and they will get the message, so I have to spend time explaining what actually happened.

Reviewer is really not paying attention.