About as clean as it gets
This is the reassuring one. India has no inheritance tax and no gift tax, and the income-tax law exempts anything you receive under a will, by inheritance, or as a gift from a relative, under the proviso to Section 56(2)(x), so the Indian receipt is tax-free. New Zealand matches that and goes further: it abolished its estate duty in 1992 and its gift duty in 2011, and it has no general capital gains tax at all. So as a New Zealand resident, you are not taxed on receiving an Indian inheritance, and unlike Australia or Canada, you generally are not taxed on selling the inherited asset later either, because there is simply no capital gains tax to apply to most gains.
That makes New Zealand the cleanest of the countries an NRI is likely to live in for this purpose. You do not declare the inheritance as income, and a long-held Indian property inherited and later sold usually produces no New Zealand tax at all. There is only one narrow thing to be aware of, and it concerns selling a residential property quickly.
The one thing to watch, and the new-migrant shelter
The narrow exception is the bright-line test. New Zealand does tax the gain on residential land sold within a set period, currently two years from purchase, and this test can reach overseas residential property held by a New Zealand tax resident, so in principle a quick resale of an inherited Indian residential property could be caught. The saving grace is that inherited property is specifically excluded from the bright-line test, including overseas property, so even if you sell an inherited Indian flat soon after receiving it, the exclusion generally takes it out of the charge. In practice, then, an inherited Indian property is outside New Zealand tax on sale whether you hold it or sell it, which is why this is the gentlest case of the four.
There is also a helpful rule for new arrivals. A new migrant, or a returning New Zealander who has been away long enough, is a transitional resident for up to four years, and during that window foreign income is exempt, so early Indian rent or investment income on the inherited asset is sheltered. Once that window ends, Indian rent on the property becomes taxable in New Zealand, with a credit for the Indian tax deducted. There is no New Zealand-India inheritance treaty, but none is needed. Our role is the Indian end: the legal heir paperwork, a valuation of the Indian asset, a certificate confirming no Indian death tax, and repatriation of the inherited funds to you.