There can be a few things going on here. If your chimney flue is lined then any falling debris is safely contained and won’t be hot or burning if it falls on the register plate. It is most likely to be parging which is the old dry cement that used to line the chimney originally. This does not indicate anything is wrong structurally and happens a lot.
If the debris fell within the flexible liner, it would be sooty/ crystalised tar that landed on the top of the throat plate of the stove, ready to be re-burnt. This is fairly normal and nothing to worry about unless it happens a lot, in which case could indicate that you are burning the fuels too low and not achieving full combustion. A stove thermometer would help (if you don’t already have one).
Also talk about what could happen if you don’t have a liner and WB
Mention perfectly normal in open fire