In moving over the media, start with all the bio-media and if it all fits, you can replace the mech media. If all the bio-media doesn't fit, I tend to put it in the tank but away from the filter in or outlets. You want it to be able to function, but not as well as what is in the filter. After two weeks you can begin to remove it. The less you have in the water, the more you can remove in one go.
When the new filter is set up you can add ammonia sufficient to produce 2 ppm in your tank, Use the ammonia calculator on this site to determine how much ammonia this is and don't forget to subtract 10-15% of the supposed tank volume to allow for glass, decor, substrate etc. Wait 24 hours and test. If its at 0/0 and you can see nitrates, its ready for the fish. If not, just finish up the fishless cycle according to the directions on this site. Consider yourself at the point between the snack dose and the end of the cycle and that your 2 ppm addition is the first dose after the snack. Work with 2 ppm instead of 3 unless this is an African cichlid tank, in which case leave it at 3. it should not be more than a couple of days to get it up to full capacity.
I am not a fan of running two filters side by side to get the new one going. The reason is that the bacteria will always be found in the greatest numbers wherever in one's tanks it is the most favorable for them. For many this is in the bio-media in the filter. When setting up a new filter the bacteria in the old one don't just pack and move to the new one. What happens is that some of them are always motile and frequently flow can break pieces of the biofilm off. They will have a decent chance for some to end up in the new filter. Now what we want is for them to reproduce there, we want more bacteria in the new media than in the old. In the case of moving over the media but not all of it, we need to get the rest of it to end up in the new filter somehow.
While our bacterial colonies may be stable in terms of the number of bacteria there over time, the individual bacteria are constantly dividing and dying on a regular basis.So while the total colony functions the same, the individuals are constantly changing. And what causes more to die than are replaced in any one location is the reverse of what causes them to reproduce- too little vs excess nutrients- namely ammonia and nitrite. We have seeded the new filter by moving over media but have more media left behind.
By putting the extra old media in a less hospitable location it will still function some but it will cause the numbers in it to decrease and thus encourage those in the new media to increase. Even letting it float helps because the bacteria are photosensitive and that will help reduce their numbers. But if we leave them in the old filter and put it on the new tank, we are providing a much better environment for them in the old media and that will slow or halt their numbers from shrinking which also means there will be less reproduction going in the new media.
Just one man's opinion here. But this is pretty much how I have replaced my bio-media over the years when I have had to and I have never had issues.
Oh yes this method works equally well whether you are replacing a filter or just the media only because it has clogged beyond cleaning or is falling apart.