Selecting a smartphone has become increasingly complex due to the differences in 4G frequencies used across the world. A phone with great specification and 4G support may work perfectly in one country and not at all in another. This situation reminds me a bit the compatibility problems of analog TV with PAL, SECAM and NTSC.
I am going to explain here how to process in order to make sure that a phone purchased online or in another country works for you. First you need to find out which frequency and which type of LTE technology your telecommunication provider is using:
The frequency bands actually include the technology information. Bands 33 to 44 are implicitely assigned to TDD. But sometimes, telephone vendors only mention the frequency (ex; 2100 MHz). It is thus essential to find out if the smartphone supports FDD (band 1 or 65) or TDD (band 33 or 34) or both.
Wikipedia maintains a good page of LTE networks worldwide: https://en.wikipedia.org/wiki/List_of_LTE_networks. With this page, you can quickly find out which frequency and technology your telecommunication company is using in your country. It is also useful to find out which frequencies and technologies are used by its roaming partners abroad.
Then, if you are interested in a phone, go to http://www.gsmarena.com and search for the phone model. Here is for example a comparison between four phones:
What is important to notice in this comparison is the importance of the phone model. For example, the Huawei P9 phone has three models (EVA-L09, EVA-L19, EVA-L29). Each model supports different frequencies. Model EVA-L09 supports the widest range of frequencies. Some phones such as Leagoo support even fewer frequencies and only only one mode (FDD).
A Leagoo Elite 5 phone will thus not work in China on China Mobile 4G network (band 39, 40 and 41) but work fine in most European countries. A Huawei P9 (model EVA-L19) may not work on T-Mobile 4G network in the USA. Model EVA-L09 of Huawei P9 would however work mostly anywhere.
If you plan to stop using your smartphone after one year, any device with relevant frequencies might be suitable. But if you plan to keep your phone many years, then make sure that the smartphone manufacturer provides frequent updates of its operating system and fixes security issues quickly. Very few manufacturers actually do this: Apple and Google are probably among the best at releasing frequent security updates. Fairphone even provides access to nearly all source code of its operating system which could also be useful for security purposes. For other manufacturers, the situation might varry from one model to another.
Last but not least, you may also have a look at HTML5 support of the native operating system (html5test.com). Not all phones are equal on this aspect either.