It does!
Although they're fictional, the cavalry rescues in movies like the 1950 film Ambush and the 1953 movie The Charge at Feather River,* capture the essence of some very real rescue by cavalry troopers.
*Note: this movie is also the he source of the now-famous "Wilhelm Scream" sound effect.
During the Indian Wars, the U.S. Cavalry came to the rescue of beleaguered soldiers and civilians on several occasions. With their superior mobility, they were the best men for the fast-moving patrols or relief columns that could mount a rescue.
Of course, the a cavalrymen weren't the only men on horses. During the 1877 Nez Perce War, some infantrymen of the 5th Infantry Regiment captured ponies and rode as mounted infantry. Some wag called the mounted infantry the “11th Cavalry” (the Army only had ten cavalry regiments at the time).
Here are a few examples of cavalry rescues:
Randall's Fight (September 15th, 1867): On this brisk day, Private John Randall from G Troop, 10th Cavalry Regiment was out escorting two civilians on a hunting trip near some newly-laid railroad tracks. Like the other enlisted men of the 10th Cavalry, Randall was a black soldier. And like the other soldiers on the frontier, he was a long way from civilization. The nearest big army post, Fort Hay, Kansas, was over 45 miles away. If things went wrong, Randall was more or less on his own. And things did go wrong.
Without warning, 70 Cheyenne warriors swept down on the little party, killing both of the hunters almost instantly. Armed with nothing more than his pistol, Randall dived under a washout beneath the railroad tracks. After a fierce fight, Randall was shot in the shoulder and stabbed eleven times by spears. The Cheyenne then tried to bury him alive by caving in the hole he was sheltering in. His torment was only ended when the Cheyenne warriors departed to attack a nearby railroad camp and its tiny garrison.
Seeing the incoming Cheyenne, Sergeant Ed Davis and his nine troopers from G Troop counter-attacked on foot. Their quick action saved the camp and two civilian ox-drivers who'd had the bad luck to approach the camp just as it was being assaulted. Having driven off the Cheyenne, Davis and his men went in search of Randall, who they found barely alive. Randall and the other troopers had killed or wounded 13 Cheyenne warriors. Randall was later cited by the Army for his heroism.
This skirmish is often cited as the source of the "Buffalo Soldiers" nickname Indians gave to black soldiers like Randall. However, there's little evidence to support this claim.
Battle of Beecher Island (September 17th-25th, 1868): Major George A. Forsyth and Lieutenant Fredrick H. Beecher lead a mounted party of 48 other men to scout for Indian warriors near what is now the Colorado-Nebraska border. Forsyth's party was ambushed and trapped on a sandbar in the middle of the Arikaree River. The scouts shot their horses and used their bodies to cover their shallow, sandy rifle pits. Although they were able to hold off the hundreds of attacking Indians, Forsyth's men were in dire straits. Indian sniping killed four men and wounded fifteen more. Food and ammunition was running. So, two of Forsyth's men volunteered to go for aid and managed to reach Fort Wallace and get help.
After an cross-country epic journey dodging Indians and living on starvation rations, the two messengers staggered into a U.S. Army post. Three columns from the 10th Cavalry (the famous Buffalo Soldiers), the 2nd Cavalry, and the 10th Infantry, were sent to find and rescue Forsyth's men. Eventually, Lieutenant Colonel Louis H. Carpenter and H & I Troops of the 10th Cavalry found and relieved Forsyth's trapped men, after riding a gruelling hundred miles in a little more than a day.
Forsyth's small band had survived nine days on the sandbar, eating rotting horse meat, drinking dirty river water, and surrounded by clouds of flies buzzing around putrid corpses and horses. Beecher was dead and Forsyth was on death's door and only just survived. The chapter, "Island of Rotting Horses," in Michael Blake's Indian Yell gives a vivid description of the battle.
10th Cavalry Trooper Reuben Waller later recalled:
"Jack Stillwell [one of the two men who'd gone for help] brought us word of the fix that Beecher was in and we entered the race for the island, and in twenty-six hours, Colonel Carpenter and myself, as his hostler, rode into the rifle pits."
Mill Creek Fight (September 29th, 1879): In the Colorado Territory, a party of the 4th Infantry and the 5th Cavalry lead by Major Thomas Thornburgh was trapped by 300-400 Ute warriors. Thornburgh was killed, the mules and horses had been killed by the Ute fusillade, and the threat of Ute gunfire made it impossible for soldiers to reach the river to get water. The senior surviving officer, Captain J. Scott Payne, sent four volunteers to get help.
Luckily, one of the messengers ran into D Company, 9th Cavalry, a force of black troopers lead by Captain Francis Dodge on October 1st. Racing to the aid of Payne's trapped men, the troopers reached the scene the next day. D Company raced a 600 yards under heavy fire to reinforce the other trapped soldiers.
Finally, on October 5th, regimental commander Colonel Wesley Merritt and 500 men of the 5th Cavalry arrived, with bugles blaring the officer's call. Five more companies of the 9th Cavalry under Colonel Edward Hatch soon joined them, along with Colonel Ronald S. Mackenzie and six companies of his 4th Cavalry. All in all, 1,500 men arrived to relieve Payne, who had managed to hold out for eight bitter days, In the fighting, eleven 5th Cavalry troopers earned the Medal of Honor (which was given out much more commonly in the 1800s than it is today).
Drexel Mission Fight (December 30th, 1890): The day after the Wounded Knee Massacre, troopers of the 7th Cavalry lead by Colonel James W. Forsyth (no relation to the other luckless Forsyth, as far as I know) clashed with Lakota Indians. Forsyth's men had taken part in the Wounded Knee massacre the previous day. Now, they were looking for the Lakota warriors had attacked the 9th Cavalry's supply wagons the previous day. There was also a rumor they'd burned down the Catholic Drexel Mission. Forsyth's eight troops (a company-sized cavalry unit) and their small detachment of artillery quickly bit off more than they could chew. Black Elk's Lakota warriors were ready and eager for a fight. They greeted the 7th Cavalry with a hail of fire.
As the surprised 7th Cavalry tried to break contact with the Indians, they got trapped in a valley near White Clay Creek later known as the "Bloody Pocket." Forsyth called for help from the 9th Cavalry and their call was answered by a battalion of black troopers lead by Major Guy V. Henry, the wryly-named "Henry's Brunettes." Their arrival with three pack howitzers helped drive off the attacking Lakota. The troopers of the 9th Cavalry had ridden 100 miles and taken part in two fights within 30 hours. Three men of the 7th Cavalry would win the Medal of Honor in the Drexel Mission fight.
Private W.H. Prather of I Troop, 9th Cavalry later wrote a song about the battle:
The 9th marched out with splendid cheer the Bad Lands to explo'r --
With Colonel Henry at their head, they never fear the foe;
So they rode from Xmas even 'til dawn of Xmas day;
They 9th is of courage bold, both officers and men,
But bad luck seems to follow them and twice has took them in;
They came in contact with Big Foot's warriors in their fierce might
This chief made sure had a chance of vantage in the fight