If your horse is registered with a participating registry, you must declare him with the USDF All Breeds Awards Program with that registry. There is a form plus you send a copy of his papers. Besides earning the minimum number of scores at rated shows, you MUST read the breed registry's awards rules, as they vary and can disqualify you in you do not follow rules. Many have their own awards fee, or require the owner to join the registry, etc.
You are asking about USEF, not USDF awards, right? Those horses are earning USEF points showing at Friesian shows - I think their show circuit is run by IFSHA. Your horse has to be at least 50% Friesian and you have to be an IFSHA member, showing at IFSHA shows to earn points. It is kind of like half-Arabians showing on the Arabian show circuit - IAHA shows.
I don't know all the rules and details of points earned - USEF awards are so different from the USDF awards, but I know there are extra points earned for championship classes, championship shows, etc.