During the gas station restaurant scene, Christian has about a four- or five-day growth of beard. When he returns to the wagon, the growth is down to that of only about a day or so.
When Chris Horn is drinking the glass of water, his hand changes position on the glass when the camera cuts away and then comes back to him.
Christian Horn and his family are on a wagon train headed for the New Mexico Territory in 1847. The New Mexico Territory was established in 1850, on land that Mexico ceded to the United States in 1848.
Joe returns to the diner after picking up the rifle (which is now old and dried out). As he opens the door to re-enter the diner, he is carrying the rifle in his left hand but the inside shot of him coming through the door shows him carrying it with his right hand.
Alexander Fleming discovered the healing properties of a mold in a petri dish by accident in the 1920s, ~2 decades after the fictional Chris Horn had died in 1914, per the encyclopedia je picked up. Fleming was working in England at the time, but the story is about Chris Horn, Sr. But it's a great story made for TV.
This story is set in 1847, and Christian Horn is carrying a Trap Door Springfield Rifle made in 1866, Second Allin Conversion, as evidenced by the type and location of the rear sight just in front of the action and far to the rear of the first barrel band.
At the end when Chris brings back penicillin for his ill son, his wife immediately flips open the plastic cap via the tab without hesitation. Modern plastic snap cap vials weren't invented until the 20th century, and opening one would have been perplexing compared to the cork stopper or screw-top glass vials of the 19th century.