Can we work towards re-homing the dogs, or not so much?
A few dogs which otherwise would have been good candidates were too internally contaminated to re-home. Not too bad, it would have been something equivalent to a few x-rays a year if the dog slept on your bed every night. But the modal rescue dog has no radiation risk at all to worry about.
Beyond that, the workers in the exclusion zone are quite fond and protective of the stray dogs, so I think there would have been significant blowback if a significant portion of the dogs were re-homed.
They are free roaming, that meant they could migrate if they really want. ..
One thing to understand about Ukraine is that many animals are kind of like community pets. They don’t belong to anyone in particular, but they live in one area and are fed and cared for by all the people in the area. People (and even businesses!) build small shelters for the animals. The Silpo supermarket downstairs from the last apartment I lived in had its own little cat shelter fit for about six cats, and painted nicely with Silpo branding.