Here is a time check using May 28th as a start day.
Normal time flight out: afternoon
14 hours for the train ride over the Gobi Desert
1:30 pm train arrival
If there is a pit stop in China then ... brb
Don't think that there is an actual pit stop in China; but most first two-hour eps have various tasks and lots of traveling. So if we follow your schedule with teams leaving Seattle on May 28, then after they arrive in Beijing and clear customs at 11:30PM they may have to find cars/taxies and proceed to someplace between Beijing and Erlyan. Perhaps the Great Wall. But the teams have to bed down for the night to wait for opening "hours of operation" let's say 8:30AM. They do the task. Clue tells them to drive/take taxi to Erlyan and find train station and book train to Ulan Bator. Since train leaves at 10:30PM this gives them plenty of time to walk the wall and to do tasks, find clue and drive on to Erlyan.
Just some possibilities. So the first elimination is at the pit stop in or around Ulan Bator (Terelj).
By the way, just in case anyone really cares, the name of the desert is just Gobi. Gobi means desert in Mongolian, so it is redundant to say Gobi Desert.