http://www.ncatlab.org/nlab/show/base+change
Proper base change: For non-torsion sheaves, see Deninger article
Ayoub has general theorems which should specialize under realizations.
There should be some conceptual understanding of the duality between smooth and proper. Ask at MO maybe after reading SGA and Ayoub. This should explain the Hodge triangles of Deligne, and Tony’s question.
There are some general ideas related to passing from characteristic zero to characteristic p, maybe via special fibers of something over a DVR. Some of this is hinted at here: http://mathoverflow.net/questions/90837/comparison-between-singular-and-etale-cohomology-in-batyrevs-paper-on-birational. This is related to base change, maybe, and should allow for comparison between Betti numbers of different kinds.
nLab page on Base change theorems