What happens to chemical compounds that include radioactive nuclei, when those decay?

To deduce this, You have to specify the kind of decay and the nature of the "compund" is it a crystal, a small molecule in gas phase, a organic material?

Beta decay shifts the nucleus one position upward in PSE, thus any "compound" will be transformed into a cation by loss of an electron, and whre say a iodide Ion had been, there will be an Xe atom. (which will not "fit" chemically of course)

There will be some recoil in this process, which can cause the nucleus to leave its place.

The electron will ionize everything along its path, those products of ionisation can alter/ destroy the molecule (compound) where the electron was emitted.

Similar is the case of alpha, with a strong recoil and severly ionisation. The decaying nucleus is shifted two "down" in PSE. The alpha particle will stay in the crystal, if it is big enough. Think of Helium gassing out of Pechblende when heated.

For Gamma, recoil will be less, ionisation is distributed along a long path (maybe meters)

There are special cases in crystals, when the recoil is taken not by the emitting nucleus alone, but collectively ba the crystal lattice. (Mößbauer effect)

In general, radioactive decay is so energetic, that any chemical bonds/lattice forces are broken. What happens then is very complicated and not to be answered by a simple scheme.