Can anyone tell me if an individual’s seasonal score is just for the alliance he starts with in the beginning? Or can their score carry over when they join?
It carries over. I’m assuming you’re asking because of how Ultron was handled. That was an entirely new thing for that.
Unless they have changed something, if you switch alliances mid-season, your points go with you to the new alliance, there for dropping the alliance score for the alliance you are leaving.
This is what I figured. We have a pinch-hitter and was making sure everything wouldn’t get screwed up. Thanks guys
The proper term is mercing. 
That being said, whenever an end of season approaches, there are lots of movements as players want to land in a higher scoring alliance. So don’t be surprised if other members in your alliance decide to leave the home team.
Alliance season score is determined at the end of the season and is the sum of each individual player’s scores