I've always been told that, for barrel racing, starting with a horse with natural run can be more beneficial. It's often easier to teach a horse to turn than it is to teach a horse to run.
Mine is a QH with all running blood. He's light cued and takes almost no effort to extend. A walk to trot= hands moved forward and either asked with a cluck or a shift forward in body position.
Trot to lope/walk to lope= Nose tipped in, inside leg. Sometimes a cluck.
Lope to run ( I never ask for an abrupt take off)= Hands forward, shift body position forward, either kiss or squeeze legs.
For any extention of gait during a run, either a kiss or tap of the leg.
I do NOT kick. Kicking knocks the air out of a horse during a run and is completely inefficient.