Any change to the trip prior to departure is supposed to trigger the airline recalculating whether the price is still valid. I'm some cases, if the fare hasn't changed, the issuing agent might choose just to revalidate a segment (i.e., change the flight without reissuing the entire ticket). But, the agent is likely only to do this if the issuing airline is the carrier for which the segment is being changed. The issuing airline is usually not willing to touch a different airlines segments without reissuing.

Adding a stopover is going to trigger a reissue as new taxes need to be collected. And, the underlying e-ticket needs to reflect that a stopover was provided.