Well since shinyskarmory decided to revive this and Brammi is still apparently looking for challenges, I'll give one.
Every time you get a duplicate Pokemon due to the Nuzlocke-like restriction, the duplicate Pokemon must be added to your party until the next gym (at least, if you decide to keep it you can). It cannot share any moves with the first Pokemon of that species you captured ASAP. "Duplicate Pokemon" is defined as any Pokemon that is the same evolutionary line as a Pokemon you have in your party (duplicates of those you have in your box need not be used).
Hopefully this'll encourage you to try out some new movesets you wouldn't have considered before, which is always fun to do!