Welcome to Rootschat. Certainly many Irish census records, some church records, etc. were destroyed years ago but this doesn't really matter when you are trying to trace people in Ireland in early 1800s as there were few records to start with. Civil registration of births, deaths and Catholic marriages started 1864 (from 1845 for other marriages).
To trace your particular family in Ireland you will need to know their religion (to find church records- if they exist) and where they lived- parish if not actual townland (to find other records).
Have you traced your Drummonds through all available English census records to see if the recorded a more exact place of birth than 'Cavan' ?