There can be a few things going on here. If you have an open fire 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. The parging will land in your fireplace ready to swept away.
If your chimney flue is properly lined by a HETAS installer then any falling debris is safely contained within the flexible liner or clay liner blocks. It will be sooty/ crystalised tar and will be landing 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 with this (if you don’t already have one).
If there isn’t a liner there then the debris would fall on the register plate, along with other debris waiting for a spark to set it alight again. This could cause a chimney fire which could set fire to your house. This is why it is important to have your chimney sweep check that your woodburner or multifuel stove has a physical connection from the appliance to the top of the chimney. If it hasn’t then give us a call if you are local and we can quote for fixing the problem.
If the chimney isn’t swept often enough and the fuel being burnt isn’t seasoned properly the tar and soot can build up and completely block a liner as in the photo below. This could have caused serious harm to the owner but luckily we got there in time.