Red, the tests (as they are called) are set by the FEI (Federation Equestre Internationale), the governing body for international equestrian events.
There are many different tests for each of the different international levels,
which can be found here.
Dressage takes years of training for both horse and rider. Those that do not understand it think that the rider sits there and does nothing, which couldn't be further from thr truth! If it is done right it should
look like the rider does nothing, it's kind of the point!
For two of the three levels that are ridden at the Olympics (
Grand Prix and
Grand Prix Special) every rider must ride an identical test. The final round, later today, is the Freestyle to music (or Kur) where the rider is given a list of moves that must be demonstrated but in no particular order. The rider also has to arrange music to suit their test, with different tempos to match the different paces.
I think the moves that you are referring to are the "flying changes" which is a move ridden in canter. When a horse canters, if you watch, one front leg lands slightly in front of the other - this is the leading leg. When a horse canters around a bend or in a circle, the leading leg is on the inside, helping to keep the horse balanced. When a horse canters with the leading leg on the outside of a bend it is described as "counter canter".
The "flying change" is where the horse is asked to change its leading leg without breaking stride. A regular rider would usually just take the horse back to trot before asking the horse to canter again.
When done right it should look like the horse is skipping.
Other frequently seen moves are:
half-pass (the horse travels forwards and sideways at the same time, crossing his hind legs. Ridden in either trot or canter)
Passage (a very slow trot with a high knee action)
Piaffe (trotting on the spot)
Pirouettes (the horse turns a half or full circle with his hind legs remaining in more or less the same spot)
The moves for today's Freestyle
can be found here
Hope that explains some of it!