We've all heard it - that transferring established biomedia from an old aquarium to a new one instantly cycles it. Or that it does nothing. Welp, what contrasting opinions. So exactly what is the answer?
Well, it depends on what you define cycling to be to begin with.
Let's go through each of the common definitions and see how it stacks.
If you define cycling as 'establishing a robust level of nitrification, at least capable of handling 1ppm ammonia (i.e. generally a fully stocked tank)', then transferring old biomedia over can instantly cycle the tank, speed the cycle up, or do nothing at all.
Different nitrifiers are adapted to different conditions, and therefore if conditions are different between the new and old tank, then the nitrifiers may not perform as well (or in contrast, perform even better). It also depends on how much nitrifiers are transferred as well, i.e. how much biomedia is transferred.
Imagine this. Say your biomedia in a 20 gallon tank currently handles 1ppm ammonia/day via nitrification. Half can handle 0.5ppm ammonia/day via nitrification. So you take that half and put it in a 20 gallon tank, now you have half of the old biomedia in the old tank handling 0.5ppm ammonia there, and 0.5ppm in the new tank also handling 0.5ppm ammonia. So not quite cycled if you expect it to handle 1ppm ammonia a day, but definitely sped up! Of course, in this particular case, you also kinda lost some nitrification in the old tank... oops.
If we take the same situation above, but the transfer is to a 10 gallon tank instead of a 20 gallon tank, well the old biomedia can actually handle 1ppm ammonia in the new tank right away (half the tank size, so double the concentration). So you'll instantly cycle the new tank!
Of course, conditions matter. Take Nitrotoga sp. HW29 that was identified to have an optimal nitrite-oxidation pH of 6.8. At a pH of 7.5, it only performs 60% of the nitrite oxidation that it catalyzes at a pH of 6.8. So if the pH changed from 6.8 to 7.5, if it could handle 1ppm nitrite before in a 20 gallon tank, now it can only handle 0.6ppm nitrite (also in a 20 gallon tank). Coincidentally, it also performs worse at higher temperatures (than 22 degrees Celcius). So the old addage of 'higher pH and temperature is better for nitrification' is not necessarily true. It really depends on what species/strain of nitrifiers you have. A lot of 'bottled bacteria' products do contain nitrifiers adapted to higher pH/temperatures, such as the FritzZyme freshwater nitrifiers.
Long story short, if you transfer over enough nitrifiers, it will be an instant cycle. If not, it may just speed up the cycle. If conditions are too different or you transfer very little biomedia, it may effectively just be no different than cycling from scratch.
Now, some aquarists define cycling as just, 'establishing any amount of nitrification'. I do not like this definition, as you can have say just 0.01ppm ammonia converted to nitrate a day and that's nitrification but... realistically how impactful is that? Not very. But hey, if this is the definition you follow, then generally transferring any amount of biomedia over will 'instantly cycle' the new aquarium... again if this is the definition followed.
Finally, the last common definition of cycling is to 'ensure 1ppm ammonia is consumed a day'. So how is this different from the first definition? In that it makes no assumption of what organism is consuming the ammonia, whereas with the first statement the assumption is it is by nitrifiers. This is important, because there is no guarantee that the ammonia in our aquariums are consumed by nitrifiers. It could be by plants, algae, or a whole host of other organisms, including non-nitrifying bacteria and so on.
This is especially a pertinent point for Walstad aquariums, which can rely almost exclusively on plants to consume ammonia, in which case transferring over biomedia can do very little. Instead, the plants would be what needs to be transferred over.
There is another consideration. Even if you only have nitrifiers and no other organism consuming ammonia, there is no guarantee that the nitrifiers are only in your biomedia. They can be on other surfaces, and there are a lot of conflicting information on where they are most prevalent. Even as a scientist, I have not found a concrete answer yet, though there is certainly evidence suggesting that broadly, biomedia (as part of a filtration system) contains a higher concentration of nitrifiers than anything else (rocks, substrates, decorations, your aquarium glass, etc.). The reason could be that a higher volume of water being pushed through the biomedia in a filter than elsewhere, so nitrifiers in the biomedia interact with more ammonia/nitrite. Whether this is true or not is up for debate, and even if so does depend on your filter as well - surely it would be very different with an undergravel filter versus an internal filter for example.
Regardless, it is a consideration. If there are more nitrifiers elsewhere than in your biomedia, transferring your biomedia may not do much.
To conclude, there are a lot of factors that goes into the impact of transferring biomedia. Whatever the case, one thing you can always do is simply consider it to consider it no different than seeding the aquarium with any other method ('bottled bacteria', etc.) and cycle from there.