My vote goes to the becket slipping. Classic symptoms. If it was the hitch pin then the sister unison would be moving as well. Strings don't start to stretch before breaking, so that's out. Bridge pin migrating is a maybe but easy to rule out by visual inspection. Either change the string or cut the becket off and reinsert making sure there is a slight protrusion. Either way you're coming back again to tweak the tuning, probably a couple of times.
Re Fred's comments about pin flexing, totally agree with that method, we've discussed it before. It minimizes pitch movement and flexing counteracts twisting. On uprights tuning from the 2:00 position accomplishes that naturally. On a grand you would have to tune from the 7:00 position which is awkward ergonomically so a gentle pressing of the hammer handle down toward the string from the 2:00 position accomplishes the same thing.
Tightening the string without any pitch response means that the string is hanging up at the bearing points. Since the segment from tuning pin to bearing points (capo bar, countering bar in front duplex, etc) is so short it doesn't take much of an attempt to raise the pitch to push that segment past the break point. That is a common reason for breaking stings while tuning.
The issue of strings stretching can be understood in terms of elastic versus plastic deformation. Deformation occurs when materials are put under stress. Elastic deformation means that the material will return to its original form after the stress is removed. Plastic deformation means that the material has been stressed beyond the yield point where the atomic bonds of the material are altered. When the stress is removed the material does not return to its original form. All materials have a yield point: tuning pins, strings, plates, ribs, etc. In the case of strings, pulling the string to tension or even beyond the required tension to a degree does not reach the yield point where the string will undergo plastic deformation, or, re this discussion, start to stretch which would both alter the yield point and the break point (or point where deformation leads to failure). Strings will not, under normal tension, deform over time. The designed tension of strings is always under the yield point.
Things can happen to weaken the strings, for example corrosion, wire fatigue due to friction, etc. And strings can be pulled past the yield and break point in the process of tuning (or playing) for a variety of reasons. But the strings left on their own at tension below the yield point will not simply start to elongate and thin (undergo plastic deformation) leading to ultimate failure.