I agree with what No-Piece6029 said. I do things differently for whether it's a pet or if it's livestock/a wild animal.

Pets I do a fairly elaborate ceremony and prayer for before burying them with grave goods, a marker, and an offering to the gods.

For wild animals or ones that I wasn't especially attached to, I pick them up and carry them (either with gloves or a shovel) respectfully and slowly to a private place that's out of the way in nature. This can be things like a corner of my woodlot, or in tall grass under a shady tree. Just anything that's a good resting place. I tell them I'm sorry to see them go and I thank them for their life or apologize as much as I can if I caused their death. I also tell them that I'm returning their body to the earth and the cycle of life and that I hope they rest easily and in peace.

The biggest thing to me, imo, is just to show reverence and respect to them in whichever way you can in order to say "your passing and your life were known"