I can for sure tell you that Disaster Transport continues to run. Except, everyone always heads for that ride when it starts raining so the line become unbelievably long. However, most rides do shut down when it begins to rain very hard or thunderstorm. The big rides are usually the first to go down, especially when winds get too strong or lightning is predicted or spotted. I have ridden MF in the rain before, but it wasn't a downpour by any means. I know many others have ridden it in the rain as well. So, after all this talking, the best way to sum this all up is by saying that it often depends on how severe, but most rides will shut down for safety reasons. You can usually get away with rain riding when it's light, but not when it's a downpour. Hope that helps.
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and remember, if you can't be an athlete, be an athletic supporter