Python 3.8 (pre-release), 103 bytes

exec(s:="from random import*;c='e'\nwhile c:print(end=c);c=choice(f'exec(s:={s!r})'.split(c)[1:])[:1]")

Try it online!

How it works?

• I started from the quine exec(s:="print('exec(s:=%r)'%s)")

then the core program is the following :

from random import*
c='e'
while c:
  print(end=c)
  c=choice('<the quine string>'.split(c)[1:])[:1]

• I assign c to the first letter of our quine (e).

while c is not the empty string :

• I print c
• I randomly choose a successor of c beween all possible successor

to choose a successor :

• let say our string is "abacbdeb" and I want a successor of 'b':
• I split the string using 'b' as separator : ['a','ac','de','']
• I remove the first substing which corresponds to the char before the first occurence of b : ['ac','de','']
• I take up to 1 char of these substring : ['a','d','']

If the char is at the end of a string  , it will have the empty string as potential successor. It only works beacuse no char is repeated twice.

Python 3.8 (pre-release), 103 102 bytes

exec(s:="from random import*;c='e'\nwhile c:print(end=c);c=choice(f'exec(s:={s!r})'.split(c)[1:])[0]")

Try it online!

the program stops by crashing due to an IndexError. To avoid crashing, [0] must be replaced by [:1] for the cost of 1 byte

How it works?

• I started from the quine exec(s:="print('exec(s:=%r)'%s)")

then the core program is the following :

from random import*
c='e'
while c:
  print(end=c)
  c=choice('<the quine string>'.split(c)[1:])[0]

• I assign c to the first letter of our quine (e).

while c is not the empty string :

• I print c
• I randomly choose a successor of c beween all possible successor

to choose a successor :

• let say our string is "abacbdeb" and I want a successor of 'b':
• I split the string using 'b' as separator : ['a','ac','de','']
• I remove the first substing which corresponds to the char before the first occurence of b : ['ac','de','']
• I take up to 1 char of these substring : ['a','d','']

If the char is at the end of a string, it will have the empty string as potential successor and will crash the program when atempting to get its first character. It only works beacuse no char is repeated twice.