I have a broodmare that I am leasing who will return to her owner at the end of January but is not due until Mid-April. I was just wondering if the game was programed to spit the foal back into my stable once it is born?
yes 
It should find its way to you and they usually do. Every now and then a bug sends them back to the mares owner though so you just have to check and report it as a bug if that happens.
Thank you