How do I get GNU screen to read .bash_profile/.bash_rc changes?

If you want screen to always treat your shell as a login shell, and source the same files that would be read if just started a new shell normally, add the following to ~/.screenrc (or maybe ~/.byobu/.screenrc, as pointed out in the comment):

shell -$SHELL

This way, you don't need to manually tell it to source your files each time you start a new screen. Though you would have to if you just made changes and wanted those changes to be reflected in your current screen.

The documentation for this (and lots of other screen details) can be found here. Basically, shell is a command to screen telling it to run the following when it needs to create a new shell. $SHELL is the usual variable holding the path to your preferred shell. And the dash - in front of $SHELL indicates that it should be run as a login shell (which will typically mean it sources your ~/.bash_profile, etc.).

It's worth pointing out, however, that screen defaults to just inheriting most environment variables from the shell where you start screen; and a login sub-shell may alter some environment variables in unexpected ways. I ran into a situation where elements of my $PATH were basically permuted. I solved the problem thanks to this particularly excellent answer on superuser.

You may notice the source command available. It's important to note that this sources a file of screen commands, rather than shell commands. Other relevant (screen) commands include eval and exec.

You have to do it in each screen that you have open since they are all different shells. If you need the change every time a new shell is opened, I suggest you put the changes in ~/.bashrc instead.

Apparently, you can send a command to all windows at once using this syntax:

C-a :at "#" stuff "source ~/.bash_profile^M"