I am in the process of adding cross-body bags to my wardrobe for summer. I usually only carry satchel styles, but I have ruined more than one lovely bag by hooking the handles over my crooked elbow (that's "crooked" with one syllable, not two) while I have a layer of sunscreen on my forearms. Plus in our hot, humid summers, satchel handles get sticky and gross.
So, crossbody it is, but I can already anticipate problems in the bust region.
I would say "always wear a crossbody over some sort of structured jacket and never over just a light knit top" to minimize the impression that you're lassoing a breast, but who am I kidding -- in the heat of summer, the likelihood that I will be wearing a structured jacket is about zero. Even a cotton denim jacket can be unbearable in our humidity.
You might think woven button-downs would solve this problem, but I'm not convinced. Seems like a crossbody would just pull open that button gap that appears when I wear a woven button-down, no matter how many times I size up.