And the next day, Big Red came home. I went and spoke to the neighbor, he's a nice old guy. As these roosters will probably just keep walking back, I've promised him 2 other roosters who don't already know the way after he's finished working on his pen. Their wings will be clipped before they leave and they'll be penned (rather then yarded) with all the ladies for a few days to encourage them to stay.
Or, when he's ready, I may give him a few chicks. Whichever he prefers.
He swears he's seen our dog back, though he's been kept on leash, even in the house (so no chance of going out the un-screened window) since then.
Ah, well. We don't have the only beagle in the world.