Glaciers move. Snow falls neat the top, compresses into ice, slides downhill to melts when it gets warmer. That happens 24/7 no matter what. How long a body, or rock, or anything else stays in the glacier is not a function of heat so much as where it enters the ice. Glaciers, at least those on mountainsides, don't move like rivers. It's more of a U-shape path. Start near the top and you will be pulled in deeper (more snow on top of your) travel further and melt out near the bottom. Die near the mid point and you will melt out very much sooner because You won't have as much snow/ice above you before you get passed the point where melting overcomes snow accumulation.
See http://www.climatesnack.com/2013/10/04/ice-can-flow-like-ket...
Climate change may accellerate the process a little if warming causes the glacier to shrink, but the greater factor is always the point (altitude) where the body enters the ice.