To elaborate just a bit more on the vector interpretation of what's going on. Any time you have a system with a force acting upon it that is always perpendicular to the instantaneous direction of motion, a curved trajectory will result. This is, for instance, why objects in the solar system like planets, comets, and meteors orbit the sun in elliptical, circular, or hyperbolic trajectories (the gravitational pull is straight toward the center of the sun, even though most objects are inclined to move around it). This is also why when you tie a weight at the end of the string and swing it around above your head in circles, the harder and faster you oscillate your wrist the faster around the weight goes in a circle (rather than pushing the weight forward, the string tugs the weight in a new direction harder and harder). The concept is called "centripetal acceleration," and a similar effect results on a canopy any time there is a horizontal lift component directed perpendicular to the instantaneous horizontal flight path. This can be achieved by initiating a tilt about the roll axis, either via manual deformation of the wing through riser input/harness turns, or by establishing a spanwise vertical lift gradient along the wing through toggle application (causing the low-lift side of the wing to drop). While vertical motion may affect the magnitude of the lift vector, the curved horizontal motion is essentially an independent phenomenon...as long as that lift vector is tilted so that a component of it resides perpendicular to the flight path, a curved trajectory (turn) will always result.
Curiously, Brian comments in his book on how a jumper can actually temporarily speed up a toggle turn by first initiating a toggle turn, and then in mid-turn applying more brakes evenly to each side. This is because once the parachute is turning, an even application of brakes (maintaining the offset between the left and right toggles) temporarily increases the magnitude of the lift vector without changing its direction, and thus the centripetal force increases, speeding up the turn (as if the artificial "sun" pulling us in orbit suddenly gained more mass). Of course, as the airspeed slows down this effect decreases, but initially it seems like the effect could be pretty profound. "Ignorance more frequently begets confidence than knowledge." ---Charles Darwin