As much as I've ridden Amtrak ( and planes in the old days) I've seen numerous deaths and Medical Emergencies occur aboard trains!
As was said by others, normally an Ambulance and LE personnel will meet the train, talk with the OBS, if necessary pick up the body or patient, and then the train will roll on!
My most memorable occurrence aboard a LD train happened on the Sunset Ltd. in the wilds of West Texas between Del Rio and Alpine where a man in a Bedroom passed away in his room!
Since the closest town with Medical facilities was Alpine, which is a crew change and service stop, the train highballed on to Alpine where we were met by officials who removed the body while we were doing station work, those boarding were then allowed on the train and we left right on time heading West!
I would guess that if the same thing happened between Alpine and El Paso, the train would go on to El Paso since there isn't anything to speak of on the Sunset Route between Marfa and El Paso, a distance of 200 Miles!