General:
* Inheritance
* Continuation
* Sequence
* Order
* Progression
* Line
* Series
Specifically for a line of rulers or leaders:
* Dynasty
* Reigns
* Line of descent
* Heir apparent
In biology:
* Ecological succession
* Plant succession
* Seral stage
* Community development
In other contexts:
* Progression (for a series of events)
* Flow (for a continuous series of events)
* Transmission (for the passing on of something)
* Handoff (for a transfer of responsibility)
The best synonym will depend on the specific meaning you are trying to convey.