This isn't a question about the tubebuddy plugin, so I've moved it to YouTube discussion.
Normally I'd wonder what the point would be in posting a youtube live from another channel, to your own channel, but I know Nick Nimmin has done specifically this with his own channel, and having his 24/7 music stream that he has running on another channel, as a featured video on his channel.
So normally I wouldn't recommend it, but you CAN do it, but I'd consider carefully if it would work for your channel.
The other point to clear up, is that you can add another channel's livestream, to one of your own personal public playlists, and then have that playlist feature on your channel main page (this is how Nick Nimmin does it) BUT downloading someone else's livestream, and then UPLOADING it to your channel is against copyright if you didn't create the livestream, and could risk you getting a copyright strike.
So again, be careful. Personally I'd only do it if it's related.
Some channels upload livestream replays of their twitch livestreams, and then upload them to their youtube channel, this can make sense in some situations, especially due to the time limited nature of twitch vods. But again, it depends on the purpose.