This is what happened to our Belgian mare, Epona.
Her owner abandoned her, stopped paying her board and didn't have her feet trimmed for months. The BO contacted him and told him she had people interested in buying her (us) and that he owed her over a thousand dollars in board fees and that Epona's feet hadn't been trimmed in six months and were in a horrible state....and told him, sell her or else I will place a lien and do so. He opted to do the right thing and sell her and clear his debt. Sales contract stated that the purchase price was used to clear his debt to the stable owner, with the rest going to him.
So, if the horse is abandoned, BO should try to contact the owner and find a resolution. If not possible, horse becomes the property of the BO once a lien is filed.