Hi Paul,
The answer depends somewhat on what you want to do. If you intend to intend to move the entire current fish population over to the new tank and the entire biomedia material, its possible to sometimes make this move all at one time without too much mishap. In some cases people even want to shut down the old tank and don't mind just moving the old filter over to the new tank indefinately, while a new filter slowly adjusts over the next 6 months. But in other cases people want to leave their current population in the old tank and just start cycling the new filter and new tank, so the members will need to know more detail about your plan.
A fish bioload and the mature biomedia that has balanced with it can be moved as a pair. The water volume is not a significant factor. When fishless cycling, however, the water volume -is- significant because the ammonia dosing needs to reach the correct concentration to encourage the correct bacterial species. That's what the 5ppm is all about. One concern about moving a fish population and putting the old media in to a new filter is that in rare cases the bacterial colonies will somehow be disturbed and not function like they did in the old filter. This can cause a period and ammonia and nitrite(NO2) is not correctly processed and the fish are put back in danger, just as if they were in a fish-in cycling situation.
A really great way to fishless cycle a new aquarium with new filter is to use some of your mature media to "seed" the fishless cycle in a strong way and shorten the fishless cycling process. Up to 1/3 (and this is the amount I recommend) of the "biomedia" (you have to determine which media in your filter you think that is) in your mature filter can be removed, kept moist, and worked in to the biomedia area of the new filter. Its ideal to position the mature media right next to the new biomedia, just prior to the new biomedia in the direction of water flow within the filter (this means you really need to know your flow direction, not just guess at it.) You sometimes have to get creative, using scissors or whatever to accomplish this media transfer. Its much easier with big external cannister filters than it is with small proprietary filters, but it can be accomplished if you work at it. Often sets of new biomedia must be purchased for each filter type to help you with the mechanical aspects.
Once you've managed to get the 1/3 of mature media into the new filter (which should already be in a 5ppm fishless cycling tank situation before doing this process) then you should also "clean" the mature filter in the new tank. Yes, this will be messy in your new tank, but the new filter will suck up the mess eventually and it will further help the fishless cycling process. Often, with a good "seed" process, it will only take a couple of weeks to finish the fishless cycle and the qualification week on the new filter, but every case is different.
~~waterdrop~~