I consider myself to decently knowledgeable on the american revolution but i have never heard of the incident depicted in the picture. (http://thehermitage.com/wp-content/uploads/2015/05/29228.jpg) from what i can tell it depicts rebel light Dragoons charging a british infantry square of grenadiers but to my knowledge there was no such charge as there was not all that much cavalry involved in the war compared to infantry due to the wooded terrain of north america and i have never read of British regulars ever having to form square.
The incident in question is the Battle of Hanging Rock in South Carolina, which took place in August 1780. The British had taken Charles Town in May of the same year, and were pursuing what John Shy has referred to as the "southern strategy": believing the South to be the soft underbelly of Patriot resistance, they sought to reach out to Loyalists in the region, while at the same time using military force to bring a widening swath of territory under their control. The Patriots, meanwhile, tried to thwart this strategy by disrupting supply lines, staging surprise attacks, and creating an atmosphere of instability that made it impossible for the British to re-establish regular government.
The British had some early success in this campaign, as their General Banastre Tarleton chased down a force of Continentals led by Abraham Buford and defeated them decisively at the Battle of Waxhaws on May 29th. The battle consolidated British control over South Carolina, but the alleged atrocities committed by British troops at Waxhaws inspired many in the backcountry to take up arms.
Over the next couple of months, the Continentals continued to harass British outposts in the interior of the Carolinas, and the Battle of Hanging Rock was part of this campaign of destabilization. By early August General Thomas Sumter had decided upon an attack on the British encampment at Rocky Mount, with a diversionary attack at Hanging Rock. He called for reinforcements, including an independent company of cavalry led by William Richardson Davie which was sent to Hanging Rock.
The attack on the British began just after dawn, led by three mounted detachments, and two camps of British and Loyalist forces were quickly routed and forced to retreat. A third camp of the Prince of Wales Regiment managed to escape into the woods, and slowed the Continental advance with crossfire. At this point, the British formed a hollow square in the centre of the cleared ground and set up a cannon to defend themselves. It is this scene, presumably, that is depicted in the engraving.
For the record, the Continental did have a number of cavalry regiments, chiefly located in Virginia and the rest of the South: notably the 1st Continental Light Dragoons and the 3rd Light Dragoons.
For a first-hand account of the Battle of Hanging Rock, see the Notes of Gen. Richard Winn from 1780.
