A brutal and all out way of fighting. Beating someone down badly. To fight all out.
He beat his ass down Brooklyn style boy.
To toss litter (particularly food wrappers) onto the street, sidewalk or gutter while walking instead of waiting to find a trash can.
There are no trash cans around here/they are overflowing - oh well, I'll just get rid of this McD's trash brooklyn style (tosses mcd's litter in the gutter keeps walking).
The native new yorker unwrapped his bagel sandwich and got rid of the foil and paper wrapping brooklyn style. (tosses it to feet while continuing walking).