Solution
You can use a 32-bit CompactPCI module or controller in a PXI chassis, and any PXI module or controller can be used in a cPCI chassis.
A PXI module used in a cPCI chassis will not be able to route triggers and clocks through the backplane. On the other hand, a cPCI module used in a PXI chassis will not interfere with the ability of the other PXI modules to route triggers and clocks through the backplane. It will simply not be able to route those signals itself. If you need to share clocks or triggers, the communication must occur through the front panel of the module.
NOTE: It's crucial to ensure that the voltage rating of the device (3.3V or 5V) matches that of the chassis.