Exploring the Possibility of Matter Convergence Across Universes
The scenario laid out in this question is a hypothetical multiverse. Notional theories of a greater multiverse of which our own universe is a part are receiving more and more popular attention. While there is very little scientific evidence for additional universes, I wouldn’t dismiss them as fantasy just yet. The analytical constructs for a multiverse are becoming more robust, and there may eventually be some evidence for it. The question assumes additional universes, so let's answer as if there are.
The question really has two parts:
1. Is It Possible That Matter Expanding from the Big Bang Is Approaching New Singularities?
First, it asks if matter originating with our universe's Big Bang is approaching new singularities. As a technical matter, the dominant Einsteinian view is that new singularities are being formed in our universe all the time. This may not be correct as discussed in the "footnote" below. What most cosmologists believe in any case is that singularities are formed as black holes are formed. Our universe is littered with black holes. So on the face of it, the answer to part one of the question is "yes” matter in our universe is approaching new singularities as and to the extent that it is being captured by black holes.
2. Is It Possible That Matter from Our Universe Is Converging with Matter from Other Universes?
Second, the question asks whether matter from our universe is converging with matter from other universes. As far as we know and a hypothesis common to most if not all theories of a multiverse, individual universes are shut off from one another. They cannot be traversed. While this hypothesis is as yet untested, it is worth bearing in mind that the basic physical parameters of our universe do not seem to be prescribed by any cosmological condition; they are apparently arbitrary.
For example, the value of the strong force that binds subatomic particles together could have just as easily have been too weak to do that job, meaning atoms and the rest of Newtonian matter as we know it to be would never have formed. The popular explanation then for how and why our universe has the physical parameters it has is akin to trial and error. The multiverse generated enough universes with random physical parameters until our own emerged, enabling atoms, elements, molecules, amino acids, biology, and eventually intelligent life to ponder it all. The randomness of physical parameters is one of if not the primary motivator to conjuring up the multiverse. Alternative explanations would seem to require abandoning science in favor of the divine.
Taking the premise that physical parameters probably vary from universe to universe, we are tempted to do a thought experiment and ask ourselves what happens if you mix the "stuff" of a universe with one set of physical parameters with the stuff of another universe with different physical parameters. Might the parameters of one universe come to dominate those of the other, resulting in some kind of transformational annexation? Might this have already occurred, and if so, in our own universe? Evidence for this would almost certainly remain, but none has been spotted. Or might a donating universe merely infuse a recipient universe with its stuff, presenting anomalies to be discovered within the recipient? We haven't seen any evidence for this either. The tentative conclusion on this front is that universes constitute fairly impermeable containers of their stuff and that they don’t open onto or into other universes donating content.
The Nature of Multiverse Theory
This separation and premise of non-merger of universes seems to guide the very nature of multiverse theory. Multiverse theory has several variants. To understand the inter-permeability of universes, we should consider three different multiverse variants:
1. The Embedded Multiverse
In the first, the apparent 3-dimensions of a universe such as ours are embedded in a higher-level multidimensional (4 or more spatial dimensions) construct. This yields an apparently boundless 3-dimensional universe that can be traveled endlessly, not unlike a M?bius strip for a 2-dimensional traveler with the caveat that the traveler eventually returns to the starting position. In this multiverse variant, different configurations of the universe are encountered as probability generates major and minor variances, like worlds with different histories than our own or beings corresponding to you with subtle or not so subtle differences. Even in this multiverse variant, the boundless nature of our universe is merely apparent because the fundamental physical parameters do not change as you move away from and then toward your starting point.
2. The Inflationary Multiverse
In the second, inflation occurs in an existing universe. Inflation is a dramatic and cosmologically rapid expansion of space, think logarithmic expansion within mere seconds. There is strong evidence that this occurred early in our own universe's history. It would explain the smooth and even distribution of energy/matter represented in the cosmic background radiation. It is believed that inflation in and of itself creates space separate from its predecessor universe's space and thereby creates a new universe containing its own "stuff." There is no apparent reason why the physical parameters of the new universe would necessarily remain the same. They might or might not change from those of the predecessor universe.
3. The Elementary Multiverse
In the third, sometimes considered a classical version of multiverse theory, we consider the hypothetical "bulk," often depicted as a meta-cosmological "foam" from which new universes are belched into existence. In this multiverse variant, there is no connection - spatial or temporal - between universes; there are no predecessors or antecedents. This lack of connection, if you will, all but precludes common physical parameters except as might coincide with probability in a very large population or an infinite one of universes.
These three examples present one variant where physical parameters that probably don’t vary, one where they may or may not vary, and one where probability dictates that they would vary. It may be evident now that the inter-permeability of universes may depend on the multiverse variant. Nevertheless, as above, we don’t have much if any evidence of other universes. This may mean they don’t exist. Or it may mean that they are impermeable and hence merely undetectable. In either case, the movement of matter from our universe toward their singularities is seemingly impossible.
A Brief Footnote on Singularities Generally
A brief footnote on singularities generally comes from recent literature confirming that, notwithstanding decades of analysis, they continue to defy mathematical analysis. In other words, they may not exist after all. Einstein may have been wrong to believe that matter could be compressed to a point or an infinite density. Infinities don’t hold over time. Another explanation for what lies at the cores of black holes may be in order. An alternative explanation is "white holes."