It would depend on when the breakup happened.
If you had been on a partner-sponsored work visa, and were no longer with your partner when the visa ended, you would not be able to renew it. Even if you had been together for 12 months and more, and had applied for partner-sponsored Residence on that basis, if the relationship failed at that stage while the case was being processed, you would no longer be eligible. If the job you were working in was not a skilled job of the kind that would allow you to get a work visa, or apply under SMC, you would be liable to leave the country. (Strictly speaking, as soon as you moved out from with your partner, that would be the case, but it wouldn't be likely to come to light that quickly, unless anyone 'helpfully' reported the situation to the authorities.) If the job was a skilled one, INZ have in some cases been pretty helpful in issuing a work visa quite quickly to allow the person to be legal, but they have to stop working until they are covered by an appropriate visa.
If time has gone on and the incoming partner has got Residence by having given all the proofs of 12 months' and ongoing live-in partnership, that Residence is the same as Residence gained by any other route - SMC, for example - and the usual next steps apply.
http://www.immigration.govt.nz/opsmanual/30505.htm