What one needs to understand here are two things. First, the bacteria is not limited to being in a filter, it is also on other hard surfaces in a tank and in ones substrate, In planted tanks a lot is on/in the substrate. In non plante,d it is in the first 1/2 inch or so. However, the filter is often where the best concentration is. Second, the number or types of filters on a tank has nothing to do with how much bacteria is in a tank. This is determined by the bio-load in the tank. That is you will have enough bacteria in an established tank to handle any ammonia being created and not handled by live plants if present.
Adding a filter to an established tank does nothing to cause bacteria to multiply, only more ammonia does this. What the additional filter will do over time is to encourage the bacteria that are in the tank to spread out some. They do not actually pack up and move to the new filter, rather a few colonize there and because it is such a good place to live they will multiply while other bacteria in less hospitale areas will cease doing so. So while the bacteria don't actually individually move, the end result is the same as if they had.
Thus when you have "moved' some bacteria to the new filter, when you move it to a new uncycled tank, you must monitor the older tank initially to be sure there are no ammonia or nitrite problems there. Unless one has removed way to much bacteria, the established tank can usually restore the bacteria removed in short order- within a day or two.
You also have the alternative of move some of the bio-media from an established filter and putting that into the new filter on the new tank. Doing this and then proceeding with a fishless cycle (dosing ammonia) will cause the cycle to go much faster than normal. This acceleration happens for two reasons. First, some the ammonia guys are in place right away and this jumps starts the process. It means nitrite is being made right away. The second, and bigger benefit, comes from having some nitrite bacs in place. Now you no longer have to wait for nitrite to build up and then for the slower reproducing nitrite bacs to start to colonize. You already have some and this really gives the cycle another huge boost. Seeding bacteria can reduce the time for a fishless cycle from an average of 35 days to anywhere from a few days to two or three weeks. It all depends on how much seed bacteria one has to start. The more seed bacteria at the outset, the faster things go.