When a foreign DNA is introduced in to an organism, how it is maintained in the host and how it is transferred to the progeny of the organism?

Foreign gene is usually ligated to a plasmid vector and introduced in the host. As plasmid replicates, and makes multiple copies of itself, so does the foreign gene gets replicated and its copes are made. When the host organism divides, its progeny also receives the plasmid DNA containing the foreign gene.