For example, you have a product that has been run in a specific country then your CEO wants you to expand the market to another country with a different culture and language. Do you think it's wise to create a specialized product to cater to the market in the new country? if so, how does it feels to partnering to local UX team/agency/professional to help your design process? especially the user research and maybe usability testing.
If we're talking solely about UX work then there are advantages and disadvantages to having people from another culture.
I work for US companies and I've worked with agencies who've brought in consultants from around the world (Australian, Indian, Russian and Swedish). The quality of the UX work was not dependent on the country of origin.
The advantage is a different perspective - an instance where they find something to be new or different and point that out. The only disadvantage I've found are time zone constraints when they return to their home country. (It's unrealistic to ask people to relocate for 6-12 months for a project.
So, to answer your question, a good UX person can transition to another country. The one real lack, if there is one, will be the ability to help with labeling, and other copy writing based work.
Do you think it's wise to create a specialized product to cater to the market in the new country?
You have the resources to create a new product from scratch despite the language and cultural barriers? I'd think it preferable to adapt the existing product, primarily via translations. Then going forward, try to create unified experiences that work well regardless of language.
... how does it feels to partnering to local UX team/agency/professional to help your design process?
Since operating systems use pretty much the same interfaces world-wide, the main issues should be ensuring appropriate translations and understandable iconography. Having multi-lingual and native teams work on those issues should be invaluable.
I can see value in having a UX professional from the country you are planning to target, but the real need is to be sure and do UX testing with users from that country.
You could hire a UX team on site, either from your country or the other one or a mix of both. You could also conduct online/remote user testing. Either way, I do think it is important to test your product with users from the target country. Then you will not be guessing at their pain-points and goals.