Firstly, I think the Kakera world of the Higurashi universe is centered and limited to only the timeline in which Hanyuu can travel in, so, the closest (but not exact) definition that’s presented for that in Higurashi, is 1978(Himatsubushi)-June 1983. I also think that this particular Kakera world is Rika’s Kakera world, cause only she can travel to its different fragments and only entities that are related to her are seen in it, Hanyuu and Frederica Bernkastel. Also, I think her friends’ consciousness passing on in the fragments is limited only to June 1983, that’s the part of the maze that they are trapped in, so like Hanyuu and Rika who can travel to any part of their limited teritory (1978-June 1983) and keep their memories intact, so can the other characters during their limited territory (June 1983), just that their memories are much more vague, of course. In the end though, I think all the previous fragments disappear and the one that remains is the Matsuribayashi one, so you can even see the characters remembering some things from the previous worlds, as the fragments being put together and their memories coming back to them again.
As for the worlds going on in the many fragments in which Rika was murdered, I think it’s all connected to the time that Rika has spent travelling through different Fragments. Rika says that 100 years have passed since she started repeating June 1983, so these 100 years did not just pass by in Rika and Hanyuu’s minds, they probably went on in the Fragments in which Rika died, too. For example, in the very first fragment, there would have passed one hundred years, in a fragment they left a bit later than that a little less, in one they left later than that less, and the sooner they left a fragment, less time would pass. For example, in Rika and Hanyuu’s minds, Tsumihoroboshi could’ve happened 20 years ago (the amount of years that have passed when Akasaka visits Hinamizawa) and only a few days would pass in the Minagoroshi fragment, cause they would solve everything in Matsuribayashi and the whole fragment system would not be needed anymore.