When you spun them off, did you leave them with significant money?

The AI won't build branches if they're losing money and they do not have a significant cash reserve to cover its losses. The AI also only makes decisions 4 turns per year. So if you left them with significant factories, they're bleeding money 1-3 turns before making choices. When it comes time for them to make choices, they're losing enough money that expansion isn't what the AI is thinking about, instead, they're cutting the red line. It's a chicken and egg problem.

Will they buy more over time?

It depends on how much money they have, how much money they're losing, and what the AI is prioritizing.

Is there a mechanism for me to help them with this?

If they're a wholly owned subsidiary, reacquire them, assign them some branches and reasonable factories for the number of branches. Then spin them off with a decent amount of cash. Else than, spin them off with nothing but money.

Be sure to backup your save before doing anything.

Does the amount of money I put into the creation of the subsidiary influence it?

Greatly so.

Also I do not understand very well the mechanism of the branches, because if I sell branches when the subsidiary is created, I lose the network in the city of the branch and must start over my establishment.

The game was not originally designed to allow players to take over companies, nor spin them off as subsidiaries, nor give other companies (the spinnoffs) your factories/branches. These features are feature creep from extensive development without redesign of the underlying systems. The game only supports 1 branch and 1 factory per city per parent company. So it's an all or nothing thing.

So when you spin off a company and you include a branch, then yes, the subsidiary takes the entire network for that city. This is because of the limitations of the underlying system written for branches ~12 years ago.

While I don't have a bounty specifically for multiple branches in a city, Bounty #82 [www.ventdev.com] in our Feature Bounty System[www.ventdev.com] would set the framework for having multiple buildings of the same type in a city. That could probably be extended to allow multiple branches.