By default, tmux uses a 500 millisecond escape-time, which causes problems with multi-function keyboard keys1.
Set it to 0 instead:
set -s escape-time 0
For example, I have my
↩︎<caps lock>key mapped to<ctrl>on hold, and<escape>on tap. With tmux’s default escape time, tapping it without waiting for half a second before the next key press results in a<ctrl>being sent instead of an<escape>.