If you want to use "squeezings", you can just clean your old filter in the new tank. That way all the stuff on the old filter will be carried by water flow into the new filter. A part of what gets carried into the new filter will be bacteria that were dislodged from the old filter media and will now take up residence in the new filter. When I do this, my new filter is usually ready to use within a week instead of 4 to 6 weeks with a more traditional fishless cycle. Of course, the new filter's bacteria need feeding from the beginning so as soon as I am done cleaning the old filter I add my first ammonia dose. I don't want to do that early because the old filter will be going back onto the old tank and I don't want to dose my fish with any ammonia, not even that bit that would end up in the cleaned filter media.
As Truck said, another way to clone a filter is to remove some of the filter media, the cleaning stuff inside the filter, from an old filter and put it into the new filter. The bacteria that is in the old filter will then be able to get a foothold in the media of the new filter. This can be more difficult to do successfully because filters can vary so much in what media they have in them and how they are able to hold each other's filter media.