The compositions of planet-forming disks are set by a combination of material inherited from the interstellar medium and material reprocessed during disk formation and evolution. Indeed, comets and primitive meteorites exhibit interstellar-like isotopic ratios and/or volatile compositions, supporting that some pristine material was incorporated intact into icy planetesimals in the solar nebula. To date, the survival of volatile interstellar material in the disk stage has not been modeled using realistic disk physics. Here, we present a modeling framework to track the destruction of interstellar ices on dust grains undergoing transport processes within a disk, with a particular focus on explaining the incorporation of pristine material into icy planetesimals. We find that it is difficult to explain inheritance through the local assembly of comets, as ice destruction is rapid for small (<10 μm) grains in the inner few tens of au. Instead, a plausible pathway to inheritance is to form pebbles at larger disk radii, which then drift inward to the comet-forming zone with their ices mostly preserved. Small grains beyond ~100 au can experience ice photodissociation at the tens of percent level; however, little of the ice is actually lost from the grain, likely making this a robust site for in situ ice chemistry. Our models also indicate that many complex organic species should survive passage through the disk intact. This raises the possibility that organics synthesized in the interstellar medium can be delivered to terrestrial planets by icy-body impact and thus potentially participate in origins of life chemistry.