Nothing in our current understanding of cosmology allows for matter from one universe to move to another.
Moreover, "when" and "where" are not concepts that exist in the context of extrauniversal "space". Space and time are constrained by the boundaries of our universe. "Outside" does not have a meaningful definition, because the "direction" to that universe cannot be expressed in terms related to our own universe. So even assuming you could get "outside" our universe without faster-than-light travel (there is no known way this could happen), there is then no way to express which "direction" you'd go to find the new one, because there is no space.
Even cause and effect don't hold sway outside of our universe, so it's likely impossible to "detect" a "new" universe inflation, because its temporal dimension(s) are completely unrelated to our own.
The only mechanism that offers some possibilities is string theory which is waaay beyond my pay grade, but which has been mined by science fiction authors as a means of connecting universes (Marvel did it most recently, but Crichton did it as well). It postulates that you might be able to visit other universes in the multiverse by travelling along one of potentially many other dimensions that are curled up in spaces smaller than the Planck Length. How one goes about translating along such a dimension is where the handwavium comes in, because, to put it simply, we don't know, and we don't even know if those dimensions are there.
So with the
reality check tag in place, my answer at the top stands - there's almost certainly no way to travel to a different universe; and even if there were, its inflation state is orthogonal to our own temporal dimension, so if you have a travel mechanism, you can likely arrive anywhen in the target universe. (Which violates causality in that universe, of course, but that may or may not be a problem.)
Edit in light of new information in comments and edit to the OP:
As indicated in the comments, nothing we know about cosmology suggests that a new universe could appear within an existing universe. Among other things, for a universe to begin inside our universe would violate the law of conservation of mass/energy. The singularity preceding the new Big Bang would require a universe's worth of matter and energy. Our universe's creation does not violate that law, because the law only holds sway within the boundaries of our universe.
The closest we might ever see is False vacuum collapse - a different set of laws expanding from a nucleation point at the speed of light. This could be harmless or intensely destructive, but in either case, inescapable.
Moreover, without FTL, not only is it impossible for your hypothetical species to outrun that boundary, they are not likely to be able to explore much of it (or the existing universe), assuming they survive the transition.
Edited in response to the second edit:
Notwithstanding my preceding stuff re: a new universe not forming within the confines of an existing universe, assuming that this new universe follows the pattern of our own, it'll only get down to the balmy temperature of 4000 K (3727 C) after about four hundred thousand years. So it would be a bubble eight hundred thousand lightyears across. Without FTL, that is an inconceivably huge bubble, which is still expanding outwards at the speed of light. And you'd only be seeing the first atoms being formed "inside" it.
So, still no.