I’ve got a ton of cherries and a few plums that are past their prime and have gone pretty soft. I’d like to do something with them. What can I do? I was thinking some kind of upside down cake, maybe, but I don’t even know where to start. I can make a fruit crisp, but I’m scared it’ll turn out like warm jam with a crumbly crust on top, since the fruit’s starting out a little squishy.
You say that like there’s something wrong with it. 
I think a fruit crisp would be good, even with squishy fruit. Plums would make an excellent fruit crisp.
adds plums to grocery list
Actually I had this same problem a few days ago, with squishy apricots. I made a sauce to go with a duck breast. I tossed the fruit in a pan, along with red wine, balsamic vinegar, sherry vinegar, rice vinegar, soy sauce, brown sugar, diced shallot, ginger, garlic, and rosemary. I stirred on medium heat for about 20 minutes. It turned out really nice; fruity, sweet-but-not-too-sweet, very morish.
You can toss the cherries (pitted and de-stemmed) into a saucepan with some sugar and cook for a while, then serve over vanilla ice cream.
Blender+ice+rum+squishy fruit+confectioner’s sugar = adult smoothies!
Compost.
I was going to suggest a simpler version of **otorophile’s **post. Just gently cook the fruit down with a little sugar, stirring often. Cool and refrigerate.
You can either put it into a little squeeze bottle and pipe it over ice cream or other desserts, or just chill it for a couple of days until you can come up with something better! 
I see no reason you can’t cook all the fruit together, but if you feel like dirtying two pots you could make a separate cherry sauce and a separate plum sauce - then see which one you like more!
Pit them and freeze for later use in fruit smoothies.
Like otorophile said, a sauce will work well with fruit. A simple bechamel sauce works well with fruit and a little garlic added. Served over orange roughy, mahi mahi or other mild white fish, or even chicken… yum. Sweet and savory.