I am assuming, you are talking about two people only and know that this should have been done a while ago, so will be less picky. I checked united for you, I can see saver level business to RIO on 12/24 in the afternoon, but standard level returns, assuming you are looking for 5-10 days, as your post does not mention how long you wish to stay. I checked BOM, but saw nothing. If you have status or UA Explorer, you should be able to force Standard awards if there are seats available.

For AA to RIO, only thing available is standard first. For BOM, there are business saver for Dec. 30, 31 ,6,8 and business saver returns on 1/15. For FC, go out on 1/06/14 and return 1/13/2014. Could mix and match, but nothing involving the week of Christmas.

I would post actual mileage account balances as many newbies have no idea how many miles it takes to do the trip they want. It makes no sense to look for things only to find out the OP has insufficient miles. If you are stuck on 12/25, then I would bet that standard awards may be a good idea for you, although a terrible idea in general.