It is more a question of where they were made. Martini-Henry rifles and their sporting cousins were made in Birmingham, England. The Henry part refers to the style of rifling chosen to reduce the black powder, and which later proved not to be good for nitro powders.
Francotte were a Belgian company who also made different sizes of Martini actions as well as complete rifles. They developed the variant where the complete action is mounted on a block that drops out at the twist of a lever, usually combined with a hole at the rear of the action which allows the rifle to be cleaned from the breech end.

BSA later adopted this feature for their .22 rifles.