Thanks for pointing this out, Iowamom. Of course pitchers need to watch hitters in games when they are not pitching, because they will face them at some point. That's why I don't understand not sending all the pitchers to all the games.
In pro and college ball, they almost always play series. There are strategies about who pitches in what order, and how they see lineups. It's obviously also more convenient for travel, and so they have to take the whole team.
In the Northwoods and some of the other summer leagues they play 3 or 4-game series, so all the pitchers obviously have to travel to the away series.
Some summer leagues schedule single games instead of series, which I find odd. In fact, it's been bugging me all summer, and now I know why. If the non-eligible pitchers don't go to games, then when they eventually face those batters, they will not have seen them if they weren't at the away game. It seems different from the way the rest of baseball operates.