If one or both of the cars are quite old, the dealer is likely to be offloading them immediately to a used car company anyway. All they are really doing is faciliating you selling it to the used car place.
I "traded in" an old car and the dealer was quite open that this is what he would do and even rang the guy in front of me to get an approximate value.
So, they might do this, but the amount you will get will be less than if you sell it yourself, but your mother might be willing to do this to avoid hassle.