They used transport ships -- a lot of them -- chartered from private enterprise for the most part. The Royal Navy had a formal Transport Board a few times during the period I'm familiar with, the first established in 1689 to deal with getting supplies to the army in Ireland (though in practice most of its work was to Flanders); another established after the original was discontinued in 1702 that lasted until 1717; and the longest-lasting established in 1794 and running through 1817. The last actually absorbed the responsibilities of the Sick and Hurt board, first only prisoners of war and then the entire charge of caring for wounded seaman. (The Sick and Hurt commissioners had, by the early 1800s, suffered years of horrendous bookkeeping and had unpaid accounts adding up to £2.5 million, which is one reason the board was wound up.)

The Transport Board offered employment to naval lieutenants (as captains of transport ships), but the ships themselves were taken up from the merchant service, and had warrant officers supported on naval pay.

To answer your question about the American war specifically, the transport responsibilities were handled by charters that the Army handled itself, not necessarily through naval channels, and through competing sets of merchants. The result was inconsistent service, and an inability overall to coordinate convoys in an efficient manner or to ensure consistent resupply for the troops on land -- which of course culminated in the Battle of the Chesapeake, after which Cornwallis' besieged army were finally forced to surrender.