For a Miniature Horse foal to be registered, I know several conditions must be met. Both parents must be in the registry, and they have to be at least a certain age (I think it's 4 years old). A stallion report must have been filed with the registry at the time of the breeding, or at any rate, before the foal will be eligible (I believe there is an additional charge for a late filing, and a certain point beyond which you are just out of luck. That way, they don't have to deal with someone coming to them and saying, "oh, hey I have this 10 year old mare that I want to have registered, here's the stallion report, and here's the paperwork on the foals she's had since then, I want to get them registered, too)
The owner of the mare must be a member in good standing of the registry. If, at any time, they stop paying their dues and cease to be a member, their registrations become void. When a dues-paying member of the registry sells a horse, the registration is automatically null and void. If the person buying the mini is a member of the registry, they have to present a copy of the bill of sale and a transfer of ownership request (along with the appropriate fees, of course) to the registry. The registry will then honor the registration as belonging to the new owner. A lot of formerly registered minis become unregistered minis during this stage, because someone doesn't want to jump through the hoops and pay the fees.
A foal is never given more than temporary registration; they can't be permanently registered until they are at their mature size (which is at age 4 or 5, depending on the registry). All temporary registrations expire when the horses reach that age, and you have a limited amount of time to bring them up to permanent registration. For permanent registration, a horse must be seen by either a judge or steward of the registry, and an official measurement made, to determine whether they meet the height requirements of the registry.

The owner of the mare must be a member in good standing of the registry. If, at any time, they stop paying their dues and cease to be a member, their registrations become void. When a dues-paying member of the registry sells a horse, the registration is automatically null and void. If the person buying the mini is a member of the registry, they have to present a copy of the bill of sale and a transfer of ownership request (along with the appropriate fees, of course) to the registry. The registry will then honor the registration as belonging to the new owner. A lot of formerly registered minis become unregistered minis during this stage, because someone doesn't want to jump through the hoops and pay the fees.
A foal is never given more than temporary registration; they can't be permanently registered until they are at their mature size (which is at age 4 or 5, depending on the registry). All temporary registrations expire when the horses reach that age, and you have a limited amount of time to bring them up to permanent registration. For permanent registration, a horse must be seen by either a judge or steward of the registry, and an official measurement made, to determine whether they meet the height requirements of the registry.