Never had a problem with the full costume characters (Pooh, Mickey, etc.) as I know those have a handler with them. Or a server has hovered close by to take pictures - some families want shots with the whole family and not just the kids too.
I don't know if face characters have handlers - I think a large part of the reason the full costume ones do is to help them avoid collisions as their range of vision is limited, and that reason doesn't exist for face characters. But I haven't done a face character meal in well over 10 years (1900 Park Fare...I only have pics with Alice for face, so I don't know if the Mad Hatter wasn't there period or just not there that day) so while I don't remember Alice having a handler she may have, just less immediately obvious than Tigger's.