Relocating freshwater fish requires permission from authorities on a case-by-case basis and general permits are not issued because the effects of relocating fish can very depending on the location and species. If you wish to transfer and release fw aquatic life where it isn't already present, you need permission from DOC and it requires public notification. There's a cost involved.
Most transfers are of species that already exist at the release site, which is a simpler process and involves permission from MPI. There's no cost.
Section 26ZM of the Conservation Act is the main legislative tool, but other legislation may apply.
I have lodged applications to both agencies, but if you want to inquire further, email the relevant agency, as below:
If you're obtaining the fish from the wild, you may also need a special permit from MPI (but there are other avenues).