How do you move equipment into saddlery on Howrse?

There is no more saddlery.

You can't remove the saddles or bridles out of the Saddlery, as of that's what I know.

There is no longer a saddlery on Howrse. Instead, just go to the bottom right of your horse's page where you will find the options to equip him with tack from either the EC he is boarded at, or with...

You have to buy saddlery from the store under the 'breeding farm' page.

When your horse is old enough for tack, there will be a box that says equip.