1. serving to complete or put the finishing touches to a phase of a particular action.
  2. serving to complete a military campaign by killing or capturing any remaining enemy troops: a mopping-up operation.

Origin of mopping-up

1905–10; mop up + -ing1, used attributively Unabridged Based on the Random House Unabridged Dictionary, © Random House, Inc. 2018

Examples from the Web for mopping-up

Historical Examples of mopping-up