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.