Your loose beads are most likely your problem. What's happening is when you apply the brakes, the tire moves on the rim, thus stretching the tube at the valve stem. This is actually a relatively common problem. I used to see this all the time when I was a mechanic, especially on mountain bikes. So...you can do a couple of things, either check to make sure your valve stems are still straight, and if not deflate the tires and spin them on the rim to get them straight...or, get some tires that have a tighter bead. One other question is, it's not so much how many miles are on the bike/tires, but how old they are. If the sidewalls are really dried out, this will exacerbate the problem, as the dry sidewalls will slide more easily along the rim.