Can anyone tell me how relegation in partnership is working?
Nope, it's still a common misconception even until today.
Franchising eludes to teams buying up a spot in the league. The model is being used mostly with League of Legends and the Overwatch League.
Partnership is Riot's new system where instead of teams buying up a spot, the teams apply for it according to certain criterias and standards Riot had set. Once approved, alongside the organisation's own financial and social status, Riot also help with yearly investments, lets say.