I have cablebox plugged into  one hdmi port and roku into another.  I was having to use the source/input on remote to switch between cable and roku.  I recently  discovered a thing called CEC.  It is a menu choice on some tvs to make  supported devices work  better together.  If  I  turn CEC  on,   then If I am watching cable and want to switch to the roku I can just press a key on the roku, usually home,  and it will switch to the roku, without having to switch source first.  But in order to switch back to cable, I have to use the remote to change the source.  There may be a way to switch back but have not found it.  If I turn CEC off  then I have to use the remote to switch source to roku.
I do not know if it works the same way for the firestick.  From what I read, some tvs and devices support CEC and some do not.