Thank you for your question to History Hub! I had personally never heard the particular wrinkle involving the brine rubbing, but the story of the beating of the runaways goes back to 1866, with the testimony of an ex-Arlington enslaved person named Wesley Norris to a newspaper called the National Anti-Slavery Standard. In it, Norris lays out all the details which have become part of this standard account of the incident many argue represents the most problematic aspect of the Lee legacy. Most Lee scholars writing before recently dismissed the account offhand (many choosing to trust Lee biographer Douglas Southall Freeman in doing so), but a recent biographer dug into the incident further and found more information.
She suggests that while Lee did not personally whip the escapees, he did encourage the constable who caught them as they fled toward Pennsylvania to do so and was present during their administering. She finds no evidence, however, of the more colorful parts of the account. The beatings do fit a consistent pattern among Lee's management of his wife (Mary Anna Randolph Custis Lee)'s lands during a nearly-two-year hiatus he took from the United States Army following the death of his father-in-law (and grandson of Martha Washington), George Washington Parke Custis, who originally built the house now at the center of modern Arlington National Cemetery. Lee had grown accustomed to military discipline during his time in the service and was unprepared for the conditions he confronted with the Custis stewardship of their holdings. Conflicts ensued.
The links above contain more information re: your question and more information on Lee's legacy more generally. Feel free to reply if you have any follow up questions. Thanks again for your question and I hope this helps satisfy your curiosity!