Here we can offer you the choice of Tudor horses or Galloway nags.
Tudor horses would equip border nobility, and are suitable if you intend to use your mounted Reivers as mercenaries in other armies etc.
Galloway nags were a breed of small, stocky pony, unique to the borders.The stature of these ponies was so small that they were later used as pit ponies, they are now extinct. However in Elizabethan times 'Galloways' were the horse of choice for reivers, and could manage the long distances over hills and dales with difficult terrain, without a problem.