In most cases, yes - monuments as well.

The Canadian memorial to Vimy Ridge naturally fell into German hands when France fell in May of 1940. This caused quite a stir at the time in Canada, but the German government assured everyone that the monuments would be maintained or at the very least left alone. Amazingly this monument survived the subsequent air campaign of 1943-1944 and the following Operation Overlord and was left untouched by the fighting.

As a general rule, WWI graves were often damaged by indirect fire from both sides, rather than any ill intent or active sabotage on the part of either army.