Have you ever bothered to read the Book of Mormon? Not all the peoples who lived in the Book of Mormon era were from the middle east. Some were, obviously(pertaining to the Nephites, Lamanites, and Mulekites), but there were the occasional peoples you see for a chapter who the Nephites never knew where they were from...
So you could have a small core group (specific archaeological evidence points to the Olmecs as the Nephites) of people from the Middle East, with other various tribes from elsewhere around the world.
Also, in archaeological terms, Russia is considered asiatic... So there's a lot of mixed blood that's ALWAYS been in the Americas, just like today.