Yes they can under various scenarios, but your constitutional rights still apply and the facts of the detention, investigation, and arrest still apply. Your best bet is to contact attorney Vic Carmody of MS. Vic is a regular poster here and you could do absolutely no better than to consult with him. Either look up his profile here or on www.ncdd.com under the "Find an Attorney" link.
I am a Mississippi DUI Attorney,andyour question has incomplete information.Were you ever on base,did the military police follow you off base for the stop,did you ever admit that you had been on base? Now once on base did you submit to tests,what were they,field sobriety tests,Intoxilyzer 8000 (two blows,one test score.With more information I could give you a better answer. I have a statewide DUI practice for 31 years Please see my book Mississippi DUI Law and Practice,Thompson/West 2010.Unless you fight this case then the off-base DUI will stick.
781 Libberty rd.
I was an Air Force JAG and I am aware that the area right outside Kessler AFB is concurrent jurisdiction. That means that both the military and local law enforcement authorities can arrest you. If the SPs arrested you and you got charged, I'll bet the matter is headed to the base magistrate court system. This is a level of the local federal court. In some ways, this can be a better place to face a DUI as opposed to being in state court. But the bottom line is, you need legal representation and fast. The guy who replied earlier is a Mississippi DUI lawyer and you should probably call him ASAP.
Short answer - It Depends. This question is very difficult to answer with the limited facts that are presented. If you were on the base and the military police saw you and were in pursuit of you prior to leaving, then yes they can stop you after you leave the base. You must also consider the military police may still have jurisdiction after you drive through the gates because the gates may not be at the end of the federal property. There are just too many variables to accurately answer your question. You need to take the time and contact a DUI attorney and further discuss the exact details of your case.