Mr. Robert Finch in this novel is the nephew of the dead Miss Mary Finch, the former owner of Isabel and her sister. He appears only in the first three chapters of this gripping novel, where he is shown to be rather a greedy and unprincipled individual who visits his aunt and stays deliberately because he suspects that she is near her death. As Isabel tells us, Miss Mary wasn't even dead yet before Mr. Robert began taking her property. This of course includes Isabel and her sister, as Mr. Robert conveniently overlooks the fact that his aunt freed them before her death.
The last we see of Mr. Robert in the novel is in Chapter 3, when he sells Isabel and her sister to the Stockton family and they leave with them. Prior to this, Mr. Robert had said that he did not want to spend long at his aunt's house--presumably only long enough to dissolve her property and gain what money he was able to from it before returning to his home on Rhode Island. Therefore, after the third chapter we have no idea of what happens to him and he exits this book.