Military Police actually aren't like civilian police

Marine Corps Warfighting Publication 3-34.1: Military Police in Support of the MAGTF states:

The MP mission and capabilities include support for AT/FP operations, maneuver and mobility support operations, area security, law and order, and internment operations. 

So the United States Marine Corps has law enforcement battalions now (or, rather, again). Three of
them to be exact. The linked article talks about how

Marines have been increasingly taking on the role of a street cop along with their combat duties over the past decade in Iraq and Afghanistan, where they have been in charge of training both countries' security forces. [...] 

The war on terror has also taught troops the importance of learning how to gather intelligence, secure evidence and assist local authorities in building cases to take down criminal networks. Troops have gotten better at combing raid sites for clues to help them track insurgents. 

They also have changed their approach, realizing that marching into towns to show force alienates communities. Instead, they are being taught to fan out with interpreters to strike up conversations with truck drivers, money exchangers, cellphone sellers and others. The rapport building can net valuable information that could even alert troops about potential attacks.

According to the commander of the 1st Battalion, a Major Jan Durham, "no group of Marines is better at that kind of work than the Corps' military police, who graduate from academies just like civilian cops". I am a huge fan of the Military Police (seriously), but let's not kid ourselves: they are not like "regular police". Their main wartime functions (and therefore training time) are spent on internment ops, force protection, and route and rear-area security in high intensity conflict. They have done a bunch of mentoring to host nation police forces in Iraq, Afghanistan, Kuwait, Panama, etc. But they are not police.

Their law enforcement functions are seriously limited to maintaining the peace and preventing crime, specifically around the Uniform Code of Military Justice and Federally-owned jurisdictions. Yes, they do have different authorities in different places, but this is rare. Traffic tickets and investigations, domestic disturbances, putting drunkards in the tank, and the occasional drug bust are all most MPs do on any regular basis - outside of the special investigative services such as CID or NCIS. I am sure that most MPs would love to do what civilian cops do, but that's just not their job as it's more important for them to prepare for and execute their wartime functions. But they do not gather intelligence, conduct community policing, or bring down criminal or terrorist networks. They are not the FBI.

Even their use as mentors for host nation police forces has had problems historically. Because they do not deal with the issues civilian police do - and certainly not the issues police in current or post-conflict nations face - they have little they can provide to the host nation, outside of seasoned non-commissioned officers, warrants, and officers due to lack of experience (outside of internment operations).

This is not to bag on MPs - they are a vital element of any task force. But let's recognize what they are: they are military police, not police who happen to be military. They are not going to be a quick reaction NCIS to solve terrorist or drug cartel crimes in far reaches of the globe. Particularly with only three battalions to go around. I think forming them into battalions is a great idea (and not just because the Army does it this way) - it gives commanders a greater force, with more flexibility, to address his military police objectives. Which are force protection, area security, maneuver support, some personnel protection, some tactical site exploitation, and a whole lot of internment ops. Three battalions of Marine police are not going to suffice as the world police force, in spite of what some of the article suggests.

(And please ignore the Thompson quote at the end - I don't know what he thinks Marine MPs do differently than Marine grunts to contain threats. Escalation of force is escalation of force.)

(Also, if you can identify the little piece of USMC doctrine I helped write on this topic, I'll buy you a drink or something - but not those of you who I told about it.)