I would speed up for a pidgeon. They are twats.
I learnt a very important lesson about this once, DH was driving with me and the DCs in the car on a country lane, we came round a corner and a van was coming in the opposite direction, the driver had a Collie and a Husky in the front, the Collie was on his lap, the window was down, as we drove past, for some unfathomable reason, the Collie flew out of the window (I guess he saw something in the verge) and landed right in front of our fucking car.
I cannot praise DH's presence of mind enough that he kept straight on going and right over the dog, we couldn't have avoided it at all, it had no chance, but if I'd been driving I would have attempted to swerve and would probably have killed the lot of us.
As it was, we had a horrible incident whereby the dog went under our wheels, we ran right over it and it got caught on the towbar at the back. The owner was fucking distraught as he cradled his eviscerated pet and I ran to a nearby house to phone my vet friends to be on standby (the dog obviously died).
BUT, I am SO FUCKING GLAD my husband kept on going, and I am so angry with the owner of that dog that he drove with him on his knee with the window right down, it was all so avoidable and could have been an horrendous accident.
No, you should never stop for animals if it endangers people.