Most of Central Asia has become fairly straight foward visa-wise for a lot of people. Europeans for example can travel to four out of five Stans visa free ( Turkmenistan being the exception); British nationals need an e-visa for Tajikistan , but can travel to Kazakhstan , Kyrgyzstan and Uzbekistan visa free; American passport holders need an e-visa for Uzbekistan, but are visa free for the other three; Canadanians, Australians and Kiwis can travel to four Stans visa free; and many other nationalities can travel to at least three, if not all four, of the non-Turkmenistan Stans without any type of visa.
Turkmenistan is of course the outlier, and everyone requires a visa for Turkmenistan. The process is the same, regardless of your passport, though there are some nuances about approval rates, which are to do with nationality and also other factors.
If you are a dual passport holder it is possible to swap passports between countries, if you're crossing land borders, however obviously if you can avoid that it does make things easier. Your first priority should be choosing the correct passport for Turkmenistan, as this is the one with the most variables, however if there isn't any difference between your passports for Turkmenistan, it probably makes sense to choose the passport which is better for the other four Stans (not including Turkmenistan), and then use that for Turkmenistan.
Of course if you book a tour with us, we will help you with the process for all of these countries, whether that's just the Turkmenistan visa, or deciding which passport to use, or in case you also require visas for any of the other Stans as well.