Solution
In short:
- PXI Controllers are exclusively compatible with PXI Chassis.
- PXIe Controllers are exclusively compatible* with PXIe Chassis.
*A couple of exceptions (listed below) may apply.

PXIe-to-PXIe: Most NI PXIe Controllers are compatible with any NI PXIe chassis. Only exceptions are listed below:
Exception 1 – PXIe Chassis with Integrated Controller: The PXIe-1073, PXIe-1083, and PXIe-1090 come with an Integrated MXIe Controller, and as such, do not have a controller slot (meaning they are mechanically incompatible with any PXIe Controller available in the market).
PXI-to-PXI: All NI PXI Controllers are compatible with all NI PXI Chassis. Nonetheless, bandwidth limitations of the PCI bus (used by PXI) should be taken into account. The theoretical bandwidth for PXI Systems is 132MB/s. If more bandwidth is needed, consider upgrading to PXIe, which leverages the PCIe bus instead.
PXIe-to-PXI: PXIe and PXI use different bus architectures. A PXIe controller will not fit into a PXI chassis, nor the other way around.
Note – This does not necessarily apply to PXI Peripheral Modules. Some PXI Modules are hybrid-compatible and can be used in PXIe Chassis with PXIe Hybrid slots. Check the documentation for your module to confirm if it's hybrid-compatible.