The Everett Interpretation does not actually involve the creation of any extra universes. It only uses quantum superpositions. The idea is that when quantum systems for observer and observed interact, they evolve into a superposition of orthogonal (and hence non-interacting) states, each of which jointly consists of the observed system in a particular eigenstate and the observer seeing that single outcome. (See here for a brief outline of how that might happen.) Back in the 1970s when <PERSON> was trying to raise public awareness of <PERSON>'s proposal, he used an analogy to describe what it would look like to such a quantum observer 'from the inside' - that it would be as if each outcome happened in a separate world. Hence the alternative name: 'Many Worlds Interpretation'.
So your question about the bookkeeping of multiple universes is moot. There are no multiple universes. The particles evolve into superpositions of the alternative measurement outcomes purely locally, according to the standard rules of unitary quantum mechanics.
Regarding the operation of time across multiple 'worlds', since these 'worlds' are just superposed states in the same universe, time is the same for all of them. If we consider the electron passing through two slits simultaneously, the electron passing through one slit does not 'see' itself simultaneously passing through the other. (There is no mutual repulsion due to their charge, for example.) It is as if they were in separate universes. But for the wavefunctions to interfere as they do, the space and time they are passing through has to be the same. Both time and space have to remain aligned between the alternatives.
(This gets a lot more complicated in General Relativity, where different outcomes curve space and time in different ways, and exact alignment between alternatives is impossible. I have heard that the theory of 'superspace' was invented by <PERSON> to try to deal with this, but as Quantum Gravity is still considered an unsolved problem, I assume it didn't fully work, and nobody knows how to do it. I don't have the knowledge to say any more.)
It is not the case that "all movements are connected with quantum jumps to other universes". The classic Copenhagen Interpretation picture of quantum mechanics posits two processes: (1) smooth, reversible, unitary evolution of the unobserved wavefunction, and (2) discontinuous collapse (quantum jump) of the wavefunction to a randomly-selected eigenstate whenever it is measured. The entire point of the Everett Interpretation is to get rid of process 2. 283
There is no collapse. There are no 'quantum jumps'. It is an illusion caused by the process 1 unitary evolution into a superposition of states, none of which can see any of the others. And even this illusion does not apply to the smooth process 1 changing of the state between measurements.
However, there is a way of looking at the branching Many Worlds wavefunction, described by <PERSON> in his popular science book 'The Fabric of Reality', in which the universe at different times can be seen as alternative 'worlds' in superposition, exactly like the alternatives arising from measurement outcomes. One way to see this is to consider the start time of the universe (in the conventional picture) as a random quantum event. There is some pre-existing unstable state that spontaneously decays, like a radioactive atom, at a random time. So in the <PERSON> picture, each possible decay time is a separate outcome, all of them existing together in superposition. At the present, we have a superposition of universes that started 13 billion years ago, universes that started only 5 billion years ago, universes that are only just starting right now, and other universes that started a hundred billion years ago. The entire history of every possible history is all happening at once, but we cannot see those other times for exactly the same reason we cannot see the other measurement outcomes. The 'observation' of the time of the big bang is a quantum measurement like any other, and so all possible answers to the question "How old is the universe?" exist simultaneously. We seem to see exactly one outcome to that measurement - we experience only one moment of time - for the same reason we see only one outcome to any other quantum measurement. We're actually seeing all of them, all at once, but the parts of outselves that see each outcome don't interact with one another. 'Now' is not singular because 'I' is not singular. We are legion.
So, you could consider the passage of time to be conceptually "connected with quantum jumps to other universes" in a sense, but it's not a very useful picture for calculation. From the point of view of a quantum observer in such a universe, it is as if we were moving through time, with a memory of a sequence of past events encoded in the present state, and the future yet to be decided. 283