Since $2016$ is a multiple of $504 = 7 \times 8 \times 9$, any expression using $1, 2, 3, 4, 5, 6$ that evaluates to $4$ can just multiply those three afterwards and evaluate to $2016$.

So all the following are correct:

\begin{equation} 9 \times 8 \times 7 \times (6 - 5) \times 4 \times (3 - 2) \times 1 = 2016 \\ 9 \times 8 \times 7 \times (6 - 5) \times 4 + 3 - 2 - 1 = 2016 \\ 9 \times 8 \times 7 \times 6 \div (5 + 4) \times (3 + 2 + 1) = 2016 \end{equation}

There are probably at least fifty more variations that I didn't bother to work out.

Since 2016 is a multiple of 504 = 7 x 8 x 9, any expression using 1, 2, 3, 4, 5, 6 that evaluates to 4 can just multiply those three afterwards and evaluate to 2016.

So all the following are correct:

9 x 8 x 7 x (6 - 5) x 4 x (3 - 2) x 1 = 2016
9 x 8 x 7 x (6 - 5) x 4 + 3 - 2 - 1 = 2016
9 x 8 x 7 x 6 / (5 + 4) x (3 + 2 + 1) = 2016

There are probably at least fifty more variations that I didn't bother to work out.