1. He keeps abandoned dogs.

2. He keeps missing dogs.
3. He keeps stray dogs.
4. He keeps deserted dogs.
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#1, #2, and #3 seem to be the same. What about #4? Do you use 'deserted dogs'?

1, 3, and 4 seem to be the closest in meaning. The words 'abandoned' and 'deserted' are synonyms. 'Stray' is very close.

Missing dogs are those whose owners have reported their loss and want to get them back.