All mammals exchange DNA in this way. It is called fetal microchimerism and has bad effects as well as good. While improving disease resistance it can lower autoimmune reactions.
Until recently humans were not known to have microchimerism, though other animals such as rats did. Autopsies on middle aged deceased woman found male DNA spread throughout the brain. A 101 year old lady had DNA present even at her advanced age.http://www.adventure--australia.blogspot.com/