You can with the mouse as well. st doesn't support the OSC52 (Ms termcap) for setting the selection by this method, though I have an out-of-tree patch that does support it (also available as a branch in the Wayland port).
It depends ultimately on the terminal to support it though. Only Linux terminals that support it atm are xterm and (u)rxvt, and there is an out-of-tree patch for st that I made (though it was rejected for being too long (around 40 lines).
In the latest git builds of tmux (with the latest terminfo db from ncurses), you can have clipboards propagate from remote tmux to local tmux (nested configuration) using the new OSC52 control sequence implemented in a patch from 2 weeks ago.
Also, if your terminal emulator (rxvt, xterm on Linux and mintty on Windows) support OSC52 escape sequences for clipboard setting, this means you don't need X11 forwarding anymore, since you can set your X11 clipboard from tmux.
If anyone can help me get in touch with the libvte people, I want to patch OSC 52 support in there as well, since this is a great feature that should _not_ be relying on X11 forwarding or whatever.
Keep in mind that your terminfo has to support this, so make sure your ncurses-term package is the latest release from Jan 28.
I also made a patch for st to support OSC 52 propagation into Xorg, but it was rejected by the st maintainers because it uses base64 (as the OSC de-facto spec from xterm dictates).
The patch for st is however in a branch of the Wayland-fork of st
