Essentially, if you've forced the enemy to quit the field despite their best efforts, you are now in the pursuit phase of the campaign - the period when the cavalry are used to harry the retreating enemy. This phase continues until the cavalry is called back or the enemy rallies and reforms - something which often needs getting to either friendly territory or advantageous terrain, as it's at this point the retreating force can restore their morale as they feel "safe".
With 3,500 cavalry, I'm not sure why Howe in our timeline failed to wipe Washington. MANY times he put Washington on retreat in 1777, and at least three times he got a rout (which meant the cavalry was essentially running down people who didn't resist) and perused until nightfall. Maybe it's because the British Cavalry in the Revolutionary War were all green fresh out of training. You said something about the British Army in the revolutionary war and the English Gallop, I'm going to see how I can work that in.
In this case, however, the important question is where Washington's army is and how they're getting supplies. Any army needs to either keep moving (to use forage) or to have a steady supply line. This is one of your lines of retreat, ideally not your only one. Howe will want to create a situation where Washington's best choice is to retreat from New York colony, and where his other options involve either a pitched battle at a disadvantage - or having his supply line cut.
Washington usually had a camp with a week's worth of supplies or so, so I guess after his retreats they just went to camp, packed up, and left. Longer term, I think he was reliant on provisioning off the locals (Congress never game him enough stuff), while the British could not do the same. Well, I take that back. In the New York and Saratoga Campaign, they tried to provision off the locals and paid with pounds. Burgoyne's men sometimes got shot in response, but in New York the reception to that was better.
An example of how this worked in OTL during the American Civil War was when McClellan's supply line was cut at the start of the Seven Days. McClellan being no fool, he had a retreat path to the James worked out and executed it - which forced him away from Richmond.
If he'd stayed and fought for much longer he'd have potentially been cut off - if he'd attacked Richmond at that point he'd certainly have been cut off
OTL? "Our timeline"? I really think McClellan should have advanced on Richmond AFTER Malvern Hill. Lee could no longer threaten his supply line. However McClellan believed his spies that he was outnumbered 3 to 1, so he decided to leg it. I think his spies reversed the odds.