Why aren't there popular examples (actually I haven't seen any) of the franchise model applied to online services?
E.g. franchise of a given service for local (different language) versions or franchise for maintaining specific sections of a broad service (for example an online apparel shop offering franchise opportunities for the shoes section or the scarves section.
Franchising is necessary if you want to grow fast in a capital-intensive business. Typical example: McDonalds. Opening a new restaurant requires quite a bit of a capital and is risky. McDonalds avoids the need to have cash (or debt) to open a new restaurant and it reduces its risk. McDonalds provides services (marketing, R&D of creating new product, sourcing raw ingredients cheaply) that scale without lots of investment.
Those capital restrictions don't apply to web services. Tumblr became one of the biggest sites on the internet with a handful of employees. They simply don't need anyone's help in reaching, say, Brazilian market.
Then there's an issue of trust. To offer hypothetical Brazilian Tumblr, someone would have to have access to Tumblr's servers. This level of access is not how physical franchises work: McDonalds reveals very little (relatively speaking) to their franchises.
In the older days franchising of sorts was actually popular. For example, a game that was popular on Commodore 64 would often be licensed by some third party and published on other platforms (Atari, Amiga etc.).
There were also cases of companies localizing e.g. early word processing software, for their home markets.
But those days are gone because today a company that is successful can easily afford to do the relatively small amount of work needed to localize/translate software.
I guess white-labeling applies better to online services. You will find tons of examples of services that offer the white-labeling model. It is not exactly the same as franchising but there are definitely similarities.