If I understand you correctly, you are starting up a new filter on each tank - not starting up a new tank, right? If you have a fully cycled filter to run on the tank, and that filter is up to the job of handling all the waste of the fish in the tank - then it will continue to handle all the waste. If you are placing it on a larger tank, it will struggle for a VERY short time (a day or two) for it will not be able to get all the waste laden water pumped through (it will be diluted) often enough to get it processed as easily. The new filter will pass water through it, but has no beneficial bacteria in it to process.
Bacteria will transfer from the old filter to the new filter in a matter of a week or so, unless you have a UV unit running on your tank.
You will not kill or damage your cycle in any way by simply placing the new filter on the tank. But you can speed the transition and probably make is so you notice no bump at all in the cycle if you split the seasoned media from your old filter and put 1/2 of it in the new filter. That way, as the waste is pulled through both filters, the beneficial bacteria can process it, and, with the seed, they can grow a larger colony to populate the new media.
If you are adding new fish load as you go to a larger tank, you may notice a bump in the cycle as the bacterial colony laggs in growing to accomadate the new fish load. This should not take very long to even out, either.