Alpha-Beta pruning in chess?
What you are describing is unrelated to alpha-beta pruning.  The tendency of fixed-depth minimax searches to badly underestimate or overestimate positional scores in dynamic situations is known as the horizon effect.  The problem is due to static evaluation being used on positions unsuited to such evaluation.  Static evaluation scores a position based on things like material, pawn structure, and king mobility, properties that can be determined without extending the search tree.  But some positions are transitional, needing more search to identify their nature.

The standard way to deal with this problem is to recursively apply a quiescence function to all terminal nodes that extends the search an extra ply if there are checks, forced captures or promotions available at that node.  If no such moves are available, then the position is considered quiet and the usual static evaluation function is called.