I would like to know any traditional sources/responsa that provide lenient opinions on handling pets; touching them, petting them, moving them.
I am not asking for personal psak, FYI, and nobody should use any of the answers of this question as their personal psak either. Everyone should call their own Rav to find out how they should handle their pets on Shabbat.
The default opinion will be that they are mukza, so can't be moved. Strict, yet very common opinions will prevent one from even touching their pets. I am looking for any and all (traditional, orthodox) opinions that are more lenient than this, with sources.
Even though the gemara paskens that animals are mukza the [responsa] rules that this doesn't apply to personal pets that one keeps in the house, that one is responsible for looking after the welfare of, and allows [leniency in their handling] by their owners but not by guests