It's the same method. It uses something like expect to scrape the output of the commands, and determine that it needs to enter a password. That framework could be extended to do the same thing for git/ssh prompts on the remote machine. These tools don't currently do this, but I was just stating that it would be possible to enter the password once even when running this across multiple machines. Note that I'm not necessarily encouraging that as a solution.

For example, I know that one of them (probably Fabric) defines the sudo prompt (via the -p flag or SUDO_PROMPT environment variable) to be able to pick out a sudo prompt in the text output of the command, so that it knows it needs to respond with the password.