Not quite so easy, and it depends on state laws. The barn owner can put an agister’s lien on the horse. In my state, it won’t take effect (be perfected) until 60 days after the last board payment. The one time I had to deal with this, some 20+ years ago, we had to send a certified letter and publish a notice for 3 consecutive weeks about the lien. The deadbeat boarder tried to pay part board, claiming it applied to the delinquent funds - which could have meant starting the whole process again. It was only after the 60 days + 3 weeks of notices that we could hold a sheriff’s sale of the horse. Fortunately, the boarder paid up in full the day before that.
As barn owners, we wouldn’t have been able to keep any proceeds from a sheriff’s sale beyond the money owed for back board (and maybe expenses? I don’t recall).
If you plan to board horses, don’t tolerate missing board payments, unless you’re wealthy enough and generous enough to board horses for free. We don’t do extensive screening of clients, but we do require a security deposit, we do charge a penalty for late board payments (and that increases the later you are), and we do stay on top of this. We also - just in passing, when we review our board contract with boarders - happen to mention that we have exercised the agister’s lien.