The oldest form of moving bridge to be commonly used in the British Isles are Lifting Bridges as used in Medieval Castles. However, as has been mentioned on that page, Lifting Bridges are not the easiest of structures to design for carrying roads across rivers, or canals.
The fairly obvious alternative to lifting the bridge deck clear was to move it on the level, and the easiest way to do that is to swing, or rotate it onto the bank. Again, it was probably the canal companies that first used this design, and with the narrow channels that the narrowboats used it was a fairly easy task to design a system of gears that would allow a bridge to be hand-cranked through 90 degrees onto the canal bank. However, with wider crossings or larger bridges it was more difficult, and after a few steam-driven set ups most are now either diesel or electric power (often both to provide a back up).