Peat Ballan, why would you bother moving a heavy frame around the garden when all you need do is spread the compost dig it in and plant.
The hot beds I saw used would be set up in the new year early to give heat for things you wanted early.
I think you assume the horse manure would leech into the ground under the box, well it was usually in an unwanted spot that got light but shelter from the cold winds, easy for us with a walled garden.
The base was a covering of straw bales then raw manure then more straw and a covering of soil, you had to wait for it to get up to heat, an old thermometer came in handy. These days I would use a sand box with cables and a thermostat, same result a lot less work and more control.
Horse manure in now more usually mixed with wood chippings where we once used straw, a much better mix for rotting down than wood chip but straw is expensive and in short supply, wood chip cheaper cleaner and plentiful.
The problem with any kind of wood used in manure or as mulch it takes the Nitrogen out of the ground so not recommended, well not by me.
Frank.