Metaspace propulsion is the science of moving something - usually you - between universes. The advantages of this are clear: each universe inhabited opens up an entire cosmos worth of resources for the taking, and the geometry of a megaverse is such that the hypersphere you're stuck on the surface is is usually higher-dimensional.
This comes with a price, however, which is why you don't see every planetary civilisation hopping between dimensions to grow their economy (the E-Prime Chorus is an outlier adn should not be counted). The price is typically measured in "spiral galaxies worth of matter".
But how do you use those spiral galaxies to lift yourself up from the realm you're stuck in? There are three main ways:
"Nudge"-Style Drives[]
Nudge-style drives are the simplest type of drive, being quickly available to any civilisation with the required energy. They rely on a simple principle: gravity leaks into higher dimensions (observe that most particles are very, very light; this is not a coincidence), which means that enough gravity can push matter in these higher dimensions around.
Importantly, this includes nearby universes. Through carefully modulating gravitational fields, you can start to assemble structures located in other universes, by nudging around the matter there - done correctly, this allows the transmission of information, energy, and even matter. The very concentrated mass that is used in these drives - usually a kind of "shrapnel" of microscopic, charged black holes - is called hypermatter.
Often, this is with the aid of a receiver - some kind of agent that is capable of reading the changes in gravitational fields and amplifying them. A receiver massively lowers the energy requirements for travelling between universes, because the changes in gravitational fields needed are smaller. Hypermatter optimised for communicating with a receiver, rather than creating a huge dent in the target spacetime, is called infomatter.
This approach is, of course, greatly limited; though the mass requirements are modest, considering the multiple universes involved, travel is only possible between adjacent universes, typically in realtime where acausal effects cannot be fully exploited.
"Search"-Style Drives[]
Search-style drives are more complicated, requiring truly high-dimensional construction techniques. Essentially, search-drives construct a high-dimensional "glider" embedded on their surface of the multiverse, which travels in a straight geodesic until it hits a universe and deposits its informational payload. Because the interuniversal void is outside of time, this allows for, in theory, arbitarily distant universes to be contacted instantly.
In practise, the construction of the glider is a difficult task, since it requires projection from a low-dimensional spacetime into a high-dimensional one. The simplest ones exploit the natural splitting of a universe; when seeded properly with infomatter (on the scale of billions of light years), and then forced into a state change (say, a false vacuum collapse), the universe splits in two along a high-dimensional surface and the close proximity of alternate versions of the infomatter array at the moment of splitting allows them to interact well enough to assemble the glider. This technique is known as branching.
The number of state-changes available to a universe before it stops being capable of hosting infomatter, however, is limited; only a finite number can be performed before the universe becomes mangled. This can be avoided by setting up a "rake" on the surface of the multiverse which continally returns to the same state, releasing a glider each time; this is difficult, and requires a great many universes to be expended by branching, as well as a constant stream of universes as fuel, but when performed can continue producing gliders indefinitely. Being outside of time, the only limit is how fast universes can be fed into it. The production of a repeating rake is known as looping.
These approaches, though very effective, have a key limitation that they still rely on activities performed inside of universes; branching directly requires the infomatter to be set up, and looping still requires maintenance inside its universe. This means truly infinite growth is still out of the reach of these drives.
"Array"-Style Drives[]
Despite the mistaken beliefs of many primitive universal civilisations, the multiverse is not flat - this is a good approximation over very large scales (much larger than the typical universe cohort), it is not entirely precise; the surface of a multiverse is very, in fact infinitely, convoluted, and oscillates violently.
This means that, over very large scales, gliders and rakes can come into contact with each other via a multiversal boundary, rather than a universal boundary. This distinction is very important, because the multiversal boundary is independent of time, allowing supertasks to be performed. For an array sufficiently large, it is almost certain that some two rakes will be in contact, and they can be fitted to make use of this fact properly, including capturing and exploiting universes in a truly timeless manner via the multiversal boundary that they share. Such a network is called a supertask assembler web, or SAW.
Scaling this up is possible, though expensive - the difficulty of having two members of the SAW being in contact increases. The next major milestone is when most rakes are in contact, such that almost all of the SAW can continually perform supertasks. This accelerates the activity of the entire SAW to supertask speeds, rather than just individual rakes. The entire accelerated array of rakes is known as a bundle. The advantage of a bundle is that there is very little lead time - a supertask can be initiated at any point in time in the foundation universe, rather than in the widely-spaced occasions when a small SAW has an available pair.
Using an entire SAW array simultanously also allows for the sending of very large payloads. This trades off speed for scale; aligning a SAW takes some realtime in the foundation universes, but when performed a glider the size of a SAW - exponential amounts of universes - can be sent simultaneously. This technique is known as coherent transport.
A combination of the bundle and coherent techniques, performed properly, allows a continuum of universes to be travelled to continuously; this finally allows for truly infinite growth, civilisations occupying entire multiverses.