I have had the luck that my local dry cleaner can dry clean my RTW and designer shoes. My husband got sick on his Burberry sneakers which were rubber and cloth/textile and they got all the stains out!
If it is luxury/designer, they always request I bring in the item to evaluate if they are capable of doing it or not. If they know they can't do it, they will decline.
Not all dry cleaners can do that though.
Especially if the sneakers or shoes are white, they could accidentally use chemicals or cleaners that would make the white turn yellow. I had a bad experience once years ago with another dry cleaner that completely ruined a friend's pair of pure white sneakers. (I had previously trusted them because they cleaned my sneakers perfectly fine but they were not white.)