The Brits moved south because they had most of the North wrapped up.
Trivia time: When General Washington moved south to engage Cornwallis on the Virginia peninsula in a campaign that culminated at Yorktown, he needed to move through northern Virginia very quickly despite its nearly-impenetrable woods and terrain. He called upon his decades-old friend and fellow Burgess from Prince William County, Henry Lee II of Leesylvania, who was the father of Washington's favorite dragoon (cavalry) officer, "Light-Horse Harry" Lee, and therefore, the grandfather of General Robert E. Lee. Henry Lee, Colonel of the Prince William County militia, cleared the way by building a courduroy road through the woods (note to WhiskeyPapa and Ditto: the labor was done by white militia members, not slave labor). The roadway is still in existence and known to northern Virginians today as Telegraph Road.
No, they moved south in order to split the colonies. That is the same reason they went up the Hudson Valley; to split New England from the other rebellious colonies. If the Brits had the northern colonies "wrapped up", they would have been as secure as Canada.
Walt