To be dogmatic here for a moment, it seems to be obvious that we have no soul, no "self."  Just sit quietly and "watch" your thoughts (be mindful).  You can see that all it is is a process of one thought after another, loosely connected, sometimes branching and going in circles, sometimes starting off on a new track, sometimes influenced by an uprising emotion or memory or an external stimulus. 

There is a difference between things that.are processes (where each event occurs in sequence and pushes itself along) and material of even spiritual objects, that have tangible stasis and properties.  A wave is like our mind -- it generally pushes itself along but still is influenced by the lay of the land, by the wind, by passing objects, by interaction with other waves, and so on.  The water that is the medium of the wave moves in little circles as the wave passes, but is not the wave.

What happens at death is of course that the medium the mind has for this process -- brain we suppose -- stops functioning and decays.  What then can possible happen to mind?  It is hard to imagine it continuing elsewhere, without a medium, but of course this is the foundation idea of rebirth teachings.