Branch migration describes the ability of a DNA

DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

 strand partially paired with its complement in a duplex to extend its pairing by displacing the resident strand with which it is homologous. This usually happens when two homologous double strands form a Holliday junction

Holliday junction

A Holliday junction is a mobile junction between four strands of DNA. The structure is named after Robin Holliday, who proposed it in 1964 to account for a particular type of exchange of genetic information he observed in yeast known as homologous recombination...

 and the whole structure slides into one direction; the "branch migrates". This procedure can influence the amount of information that is transferred between two recombining partners. In E. Coli RuvA and RuvB of the RuvABC

RuvABC

RuvABC is a complex of three proteins that mediate branch migration and resolve the Holliday junction created during homologous recombination in bacteria. As such, RuvABC is critical to bacterial DNA repair....

 protein complex promote branch migration until RuvC binds and resolves the Holliday junction by cleaving.
Another definition for branch migration is the extension of a D-loop

D-loop

In molecular biology, a displacement loop or D-loop is a DNA structure where the two strands of a double-stranded DNA molecule are separated for a stretch and held apart by a third strand of DNA. The third strand has a base sequence which is complementary to one of the main strands and pairs with...

when a single stranded DNA strand invades a "recipient" duplex during recombination or repair.