OK. In short, the bacteria exist in the media, matters & materials. Not in the water itself.One cannot cycle a tank by using water. The needed bacteria are, for the most part, non-motile. The live inside a bio-film attached to hard surfaces out of the light as they are somewhat photophobic. You can use the plants themselves as they will have some of the needed bacteria living on the stems and leaves.
Substrate without plants becomes anaerobic less tha an inch deep. In fact over 1/2 inch it gets iffy. Much of the roots of plants are down in what is normally an anaerobic zone of the substrate. Some plants will transport oxygen down and out of their roots which converts some of the surrounding substrate into an aerobic zone. Once there is oxygen, the nitrifying bacteria can live there and they do.
All of this is easy to see in this paper:
Petersen, Nils Risgaard‐, Jensen, Kim, (1997), Nitrification and denitrification in the rhizosphere of the aquatic macrophyte Lobelia dortmanna L., Limnology and Oceanography, 42, doi: 10.4319/lo.1997.42.3.0529.
Nitrogen and O2 transformations were studied in sediments covered by Lobelia dortmanna L.; a combination of 15N isotope pairing and microsensor (O2, NO3−, and NH4+) techniques were used. Transformation rates and microprofiles were compared with data obtained in bare sediments. The two types of sediment were incubated in doublecompartment chambers connected to a continuous flow‐through system.
The presence of L. dortmanna profoundly influenced both the nitrification‐denitrification activity and porewater profiles of O2, NO3−, and NH4+ within the sediment. The rate of coupled nitrification‐denitrification was greater than sixfold higher in L. dortmnanna‐vegetated sediment than in bare sediment throughout the light–dark cycle. Illumination of the Lobelia sediment reduced denitrification activity by ∼30%. In contrast, this process was unaffected by light–dark shifts in the bare sediment. Oxygen microprofiles showed that O2 was released from the L. dortmanna roots to the surrounding sediment both during illumination and in darkness. This release of O2 expanded the oxic sediment volume and stimulated nitrification, shown by the high concentrations of NO3− (∼30 µM) that accumulated within the rhizosphere. Both 15N2 isotope and microsensor data showed that the root‐associated nitrification site was surrounded by two sites of denitrification above and below, and this led to a more efficient coupling between nitrification and denitrification in the Lobelia sediment than in the bare sediment.
You can read the full paper here: https://aslopubs.onlinelibrary.wiley.com/doi/epdf/10.4319/lo.1997.42.3.0529
If you want to move bacteria in soil (assuming you know where in the substrate you will find nitrifying bacteria), move the soil and not the water. But as I stated earlier, you would do better just moving in plants.