How does plant grafting work?
This article analyses the exchange of DNA during the process of grafting

Exchange of Genetic Material Between Cells in Plant Tissue Grafts - Stegemann and Bock, Science 2009

Quoting from the article:


  Although the grafted tissues fuse and establish vascular connections,
  the stock (the lower part of the graft) and scion (the upper part,
  usually supplying solely aerial parts to the graft) are thought not to
  exchange their genetic materials. But grafting (whether natural or
  assisted) provides a path for horizontal gene transfer. Gene transfer
  is confined to the graft site and no long-distance transfer may occur.
  Analyzes indicating that large DNA pieces or even entire plastid
  genomes are transferred. Only plastid genes may be transferred, no
  transfer of nuclear genes occur. Plant cells are connected via
  plasmatic bridges called plasmodesmata, but the passage of large
  macromolecules requires the action of specific plasmodesmata-widening
  proteins. Whether large DNA pieces or even entire organelles can
  travel through plasmodesmata requires further investigation.
  
  Finally, although our data demonstrate the exchange of genetic
  material between grafted plants, they do not lend support to the tenet
  of Lysenkoism that “graft hybridization” would be analogous to sexual
  hybridization. Instead, our finding that gene transfer is restricted
  to the contact zone between scion and stock indicates that the changes
  can become heritable only via lateral shoot formation from the graft
  site. However, there is some reported evidence for heritable
  alterations induced by grafting and, in light of our findings,
  these cases certainly warrant detailed molecular investigation.