To explain what I mean about transitions and impulsion being related to balancing forward and backward motion, think about those number lines they tried to teach you in school. This may be a bad illustration since I really stink at doing math with positive and negative numbers, but here I go anyway.
-4 -3 -2 -1 0 1 2 3 4
Think of the amount of energy the horse is putting into moving forward or backward. At zero the horse is not moving either direction, the positive numbers are the amount of mental energy a horse is putting into moving forward the negative numbers are the amount of energy the horse is putting into moving backward. Like in math class a horse is taught simple things first:
0+1= walk forward 2-1= transition from trot to walk 0-1=back up
until we are able to do complex things like 0+3=halt to canter transition
or a positive -2 + 2= a piaffe
In our illustration, simple addition and subtraction of forward and backward energy signifies simple transitions, but when we start applying the negative numbers things get much more complicated. The piaffe requires the horse to have as much energy going forward as backward at the same time, so the +2 balances the -2 and the horse stays at 0 trotting in one place. A half halt is a quick, momentary, addition of backward energy to your forward motion. A sliding stop is rapid addition of a lot of backward energy while the horse continues think about running forward.
Excellent riders create elaborate maneuvers by doing the "math" at a very rapid, fluid pace. Of course this is a huge over simplification of the process, but seeking elegance and lightness going back and forth on the ground leads to making complex riding much easier for your horse to learn. This skill of "shaking a rope at your horse" is a vastly underestimated tool in a horseman's tool box.