We usually hunt doves around Yuma AZ, and the last couple of years our take of Eurasians has been about 20% of the total bag in the first part of the season, depending on where we are hunting. As the season progresses, the Euros become scarce. Could be they are smarter than the others. I don't think they migrate to Mexico when the weather cools. As soon as the nights get chilly, the whitewings pack up and leave the mourning doves for us in the second season.
Regarding crop damage, the stories I hear about those guys who shoot 100 Euros in a day are usually around ag land, particularly milo fields. We have never found a huntable field with large concentrations of Euros, so I suspect the stories are coming from guys who can hunt close to farmhouses, as opposed to those of us who can't shoot within 1/4 mile of an occupied structure.