I would recommend the following:
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list item Ensure you can choose the correct currency in each language. No one will purchase something in the UK if prices are in New Zealand Dollars.
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list item Detect the country to determine currency and postage options. This way visitors will spend longer on your site, which is an important part of ranking.
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list item Mention the countries you're targeting once on each page. This called be 'free delivery in the UK, New Zealand and Australia' for example.
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list item Write specific blog posts targeting your core keywords plus one of the countries only. This way you have different landing pages for each keyword. Include plenty of links to and images of products in your blog.
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list item Consider creating 3 'Best selling items by country' pages. This way you also have unique landing pages for each country.
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list item Ensure you don't mention currencies or countries by default in your metatags if you're using cookies to change them.
I hope this helps! You can see some additional tips on https://www.indigoextra.com/blog/international-seo.