I don't think you get taxed on just bringing money in to NZ.
You do get taxed on any income generated in NZ (for example income from property rental) no matter where you live or work or what nationality you are.
You will get taxed on any income generated outside NZ if the tax department decides you are tax-resident in NZ. There's no set criteria for that but roughly it depends on your associations with NZ. And remember that any allowances like car, medical insurance, education fees for children, etc are all included in the calculation of income (it's possible and has actually happened for a tax bill to be higher than an expat's total cash income for the year).
If you appear to be going away temporarily (months or years) with every intention of returning then you'll probably be tax-resident.
If you appear to be leaving permanently or indefinitely at least, then probably not. But you're better off contacting the IRD (they may or may not give you a definitive answer on your status as a tax resident), or a professional tax accountant/lawyer who has dealings with other expat NZers.
I don't know much about trustees except to say that if the tax department decides you set them up for tax evasion purposes then you'll be in trouble. Unlikely if you've set them up some time ago and the IRD had no problems with it before you left NZ.
Some background: You can have one or more of the following statuses:
- Resident in NZ - you live there and will pay tax on all income from NZ or overseas (unless there are double taxation agreements). Nationality is not relevant to whether or not you pay tax.
- Tax-resident in NZ - you may or may not live there but the IRD wants your money and you'll pay tax on all your income from NZ or overseas (unless there are double taxation agreements). Having NZ nationality makes it more likely you get this status.
- Domiciled in NZ - roughly equivalent to your nationality. Rarely changes no matter where you live or pay tax, and is not so relevant for determining your tax liability.
Most NZers living in Dubai, if it's clear they're making Dubai their home on a long-term basis, will be domiciled in NZ, but not resident or tax-resident in NZ.
Just to finish on a bizarre note, there is AFAIK a double taxation agreement between the UAE and NZ. Meaning if you pay tax in the UAE, then you won't get taxed again in NZ on the same income. Nice to know should the UAE ever introduce income tax

.