Matthew, It sounds as if the fixings are just worn, probably as a result of being incorrectly 'opened' in the past. I have never had a problem with the Press Studs coming loose; usually the opposite.
Camelman is correct. However, I have seen studs that don't cooperate before. The easy solution is to simply have the cup and stud replaced. That shouldn't take more than a minute or two if someone has the right tool and a new stud set handy. A shoemaker, a boating supply shop or an older Morgan dealer should be able to help with that.
We often travel with the top up but sidescreens off. High speed wind can often open the first stud rear of the door. To prevent that, I changed those studs to turnstiles. That will work as well.
Lorne
