How do gene locations change during crossing over events?
It depends on the regions of sequence homology between the two chromosomes. Crossing over occurs through pairing of homologous regions. If there's a substantial stretch of chromosome without a matching homologous region on its pair, that non-homologous region should loop out and not be involved in crossing over. Crossovers will occur only in paired homologous regions.
For example, if you have a chromosome I and I' as follows:
I   ABCDEFG
I'  ACDEHFGM
They should pair as:

A-(A)
B
C-(C)
D-(D)
E-(E)
(H)
F-(F)
G-(G)
(M)

Crossovers should not happen in the B, H, or M regions lacking homology with the other chromsome. Crossovers in a homologous region will just result in a new chromosome containing the left end of one crossover partner and the right end of the other.
Using I and I' above, a crossover in region E would produce either:
ABCDEHFGM or ACDEFG
Now, if you have repeated regions of homology in one or both chromosomes, or regions inverted on one chromosome vs. the other, pairing possibilities expand rapidly and all kinds of weird things can happen (and do in real life).