Another way to move things in PCB view after pasting is to use sticky on PCB layer. Once sticky is enabled all the parts on top of it will move with the board without having to select the parts themselves. The only problem with this is if that sticky board goes under any parts you don't want it to move they will end up stuck to the board for that entire session and you will have to reload Fritzing to unstick them.
When I am making break away boards and need to move the contents of one of the boards slightly I will create individual temporary boards under said contents and use sticky to rearrange them. Once I am satisfied I delete the temporary boards and you are left with the contents exactly where you wanted them without any of the parts shifting.