A galloping horse displays a four-beat footfall pattern with four distinct beats. When on the right lead the first beat is the left hind, followed by the right hind, left fore, and right fore. There is a period of suspension when all four feet are off the ground.
Racehorses in North America typically run around turns on their left lead and the straightaways on their right lead. The reason for this is that a horse is more balanced when they lead with the leg corresponding to the direction of the turn.
Jockeys often give horses a cue to change leads (often with a flick of the wrist and/or shifting of weight), and many horses learn to change automatically. A horse will get tired if they stay on one lead for too long — this is analogous to a person switching their suitcase from one hand to another while running through the airport. Here is a closeup example of a horse changing leads.
This video shows Midnight Trace negotiating the first part of the turn on his left lead before changing to the right lead at the :23 mark (it happens quickly!).
This video is a good example of a horse changing leads in the stretch. When it starts, Awesome Vision (jockey in black and gold silks and orange saddle towel) is on his left lead (the same lead a horse is on around the turn), before switching over to the right lead at the 1:46 mark (it happens quickly!!). He is shown winning the 2013 Saratoga Sunrise Stakes.
Ring Weekend (jockey in green silks and red saddle towel) was on his left lead in this video before switching to his right lead at the 1:39 mark. He is shown winning the 2014 Saranac Stakes (G3).
In the 2014 Kentucky Derby (G1), Commanding Curve (on the far outside with jockey in black and gold silks) switched to his right lead at the top of the stretch (1:44) before switching back to the left lead nearing the finish line (2:05). He finished second to Horse of the Year California Chrome.