@oristarA,
Okay. I got it. It's slang for
my way of doing, my way of operating.
When police officers are on a beat (an area they patrol all the time) they get to know the people in the area. They also get to know just how to question certain people in certain ways to get answers, it becomes their way of operating, it becomes their "thing".
So, when the younger guy says to the older guy, "I'm going to put my thing down." it means he going to go and question the suspect
his way, using his method, his 'thing'.
The older guy says basically, "Yeah, you go do that.", and see how you do.
~~
I've never heard the expression used outside of this movie, but, as an aside, the named of the organized crime group, La Cosa Nostra, translates as "Our Thing, or Our Way."
Joe(hope that helped)Nation