If your post falls over under vibration, that's a good thing. It was too loose. Seriously. I don't know where these myths come from!

For those not in the know, the bridge on a bowed string instrument isn't glued down. It's simply held by string tension. It handles this by being cut in a strain resisting shape. The best ones are cut with a crown on both faces to prevent warping. Before any "experts" object, need to check for the slight crown on the "back" of top-end pro bridges - it's generally there, just hard to see. Inside the instrument is a piece of dowel between the top and the back. This acts as a fulcrum and also puts some of the force from string downforce into the back. The post isn't glued in, only wedged in place. Fit is very important for sound and to avoid damage.

Neglected instruments often have posts that are loose. This eventually results in distortion through lack of support. If vibration drops the post, then that's great, shows it was too loose. Better to know before there's warping of the top.

They're very strong.

Pop some foam or bubble wrap in a pannier and last the thing in. Nothing bad will happen. Use more foam and I'd ride without a case - fiddles are very very tough.