Basically the bacteria colonize and multiply where conditions are most optimal. This is pretty much a flow dependent situation. the bacteria are fixed in place for the most part so they do not g out to eat, they need delivery. However, they are not completely immobile.
Some percent of the bacteria can move (how much depends on nitrogen levels). Also as their bio-film grows it becomes more subject to the shear forces of current. How the biofilm changes in shape and thickness determines how likely it is for bit or hunks potentially to become detached, The result of all these things is that while individual bacterium do not migrate, colonies can appear to do so. This happens as a result of conditions degrading in one place and becoming more hospitable in another. Reproduction slows and stops in the first spot and picks up in the second and suddenly the bulk of the bacteria in spot 1 in now in spot 2 without migration between the two spots going on.
This can also happen inside a filter or when one adds a new filter. The total bacteria numbers do not increase however, what happens is the bacteria spread out. The only way to increase the number of bacteria in a tank is for the ammonia concentrations to increase. Therefore to get new media to be colonized, some of the older media must become less so. The reason the new media will pick up some level of bacteria is it is normally a better place to live.
Because all bio-media tends to clog to some extent over time, new media will almost always offer better potential flow through and the bacteria will take advantage of this. This process will take a couple of weeks at least. However, the more bacteria one tries to move over, the less bacteria one leaves behind. The theory is if you remove about 1/4 of the bacteria from an established tank, the ensuing spikes for ammonia and nitrite will not be huge and should not last very long.
The complication is the bacteria going into the new tank will usually not be sufficient to have it be close to fully cycled at the outset. Usually some degree of fishless cycling is needed to get the new tank completely ready.
If you cannot move part of your bio-media due to the design of the filter, your next best solution it to move some of the surface of your substrate to the new tank. There is plenty of bacteria in one's substrate despite some what one often reads to the contrary. So you can get a jump start on a fishless cycle by using substrate. Unless one has a well planted tank, the bacteria will be limited to about the top 1/2 inch of your substrate.
Finally, because you live in the USA you can easily purchase some of DrTim's One and Only Nitrifying Bacteria. This is probably the single best bacteria starter product for tanks out there. The only other one I recommend is Tetra's Safe Start, and this is only if one cannot find DrTim's. If one uses either of these products is is essential to read the directions and to follow them to the letter. The most common cause of failure with these products is user error.