You can't mix LHR / LGW Airbus pilots as they work to different agreements.
As alluded to above, anybody can bid to move to LGW from LHR. However the superior bidding system at LHR is a big draw and also T&Cs are not as good at LGW (Pay cap, reduced Flight Pay Allowance Etc).
So LGW tends to be for those who live locally and enjoy the small base community, guys chasing ever diminishing (currently) command opportunities and new recruits (FPPs).