There is a risk of auto-ignition?
Yes.
Ammonium chlorate is formed which is very happy to ignite or explode on contact with fuels, or for no reason at all.
A reputable pyrotechnics source, from memory I think it was Passfire, had an incompatibility list, and described actions that can be taken to make them safe.
For example it is quite safe to roll a sulfur containing composition over a chlorate star if there is a layer in between containing carbonates or other basic salts, so that acid cannot make it through to the chlorate.
Also, one can successfully roll Potassium nitrate over Ammonium perchlorate stars if there is a Potassium perchlorate based buffer bound with nitrocellulose in between.
However, with Ammonium salts and Chlorates, it was simply put that under no circumstances should they ever be in the same star, even with many barrier layers, it's not safe.