In iTunes Connect there is a button "TestFlight Beta Testing" in order to do just that.
I have a binary uploaded and ready. A few users are selected for testing.
This time, that button just does nothing. After tapping it, it goes green. But no invitation emails are ever sent out. If I reload the page, that button is unchecked/white. Bug or did I miss something?
Update: it seems to be working again. Nothing to see. Move on.