I have this happen all the time, until I want to screenshot it happening to bug report it then it starts behaving and moves the input above the keyboard.
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Are they implementing edge-to-edge screen on this release? Can't say for certain but I've noticed this causes problems with the keyboard on other apps too.
https://developer.android.com/develop/ui/views/layout/edge-to-edge
I just installed Firefox to test this, and do not have this issue. (I normally use Fennec or IronFox)
I found this page by searching "sign up" and trying random pages to find a form near the bottom.
The page scrolls up normally when tapping the search field at the bottom of the page. I use FUTO keyboard on Android 16.



Perhaps you have the address bar at the top? It works just fine when it is set to that position as I found out just now. So it might just be a bug that manifests when the address bar is set to be at the bottom.