We don't have a command that would activate the search field itself (like pushing F3 does by default), so that couldn't be automated after a folder change, but you could do something similar by setting up a script which runs CLI QUICKSEARCH after a folder change.
That would open a text field at the bottom of the file display for searching. It would behave the same as the search field on the toolbar, but be in a different place and use up a line of the file display while open. (It would also auto-close if nothing was typed for a few seconds, or if Esc is pushed or the mouse clicked in something else).
If that still sounds useful and something you'd want, but you need the script writing for you, please link your account and I'll write the script and post it here.