When Joshua is trying to break the horse, during his 2nd try he loses his hat. As he continues to ride it keeps showing him in the close up frames with the hat, then in the pulled back frames without the hat.
When a horse is shown bucking in the Barkley corral, you can see a "bucking strap" (which causes the horse to buck) around the horse's midsection, just forward of its back legs. When that same horse has been "broken" and calm only moments later, the bucking strap has disappeared.
When Joshua is trying to break the horse, during his second try, he looses his hat. As he continues to ride, it keeps showing him in the close up frames with the hat, then pulled back frames without the hat.
The second time Joshua is on the bucking horse, he loses the right rein. It's consistently loose and flapping on the ground during most of the second bucking sequence, but Joshua somehow regains a hold on it - impossible unless someone on the ground handed it up to him.
Joshua Watson, played by singer Lou Rawls, sings an excerpt from "Streets of Laredo," also known as "The Cowboy's Lament." This ballad was not published until 1901, almost 25 years after the context time "The Big Valley" is set in.
After one of the Barkley's ranch hands is shot, Joshua Watson takes the hand back to the Barkley ranch and Nick rides through the night to get to the Morton ranch. After a 2-3 minute fight with the Morton sons, Heath arrives at the Morton ranch saying that Watson told him Nick would be there. Unless the Barkley Ranch was 2-3 minutes from the shooting site (probably not) and/or Heath galloped the majority of the way (neither he nor his horse seemed tired) it would have been impossible for him to arrive that soon after the fight.