I must have hit some button, or typed a shortcut by accident, but my workflow has been shot in the foot by this problem. If I try to select multiple meshes and move them together by typing precise coordinates into the location section of the transform, only the mesh which was selected last moves. I've also lost the ability to deselect all using the "A" key, so there is always at least one mesh selected. I tried to restore blender's default settings but the problem persisted.
Now, however doing this results in only one mesh moving. Strangely, if I try to select meshes at the same time using circle select or border select, there is still one mesh that seems to be dominant. Also, if I had another mesh selected, and then try to border select several other meshes, not including the first one, that first mesh remains "selected" and will be the only one that moves, even though other meshes are actually selected.
I have some long-winded work-arounds that I can use for the time being, but this is really slowing me down in a very inconvenient way. If I can't figure out what to do, I'm going to have to try reinstalling blender.