Chicago to Brooklyn move: from townhouse to apt

My 3 children and I are moving from downtown Chicago to Park Slope in Brooklyn. We are downsizing significantly and will be selling many belongings, moving others to local storage and then moving our clothes, toys, kitchen ware, kitchen table/chairs, chaise lounge, dresser, glider chair, bunk bed set, ~4-5 bookcases, a rug, a few end tables and lamps to Brooklyn. I completely guessed on number of boxes--I have no idea but tried to ballpark. Our place in Chicago is a 4 story townhouse which is one story up from the ground level. This means a lot of stairs. THere are interior and exterior options but it's not the easiest. In Brooklyn, our place is ground level. We have the bottom two floors of a brownstone--so bedrooms are up one flight of stairs but everything else is ground level. In NYC, there are no parking permits or ways to have the street cleared, and we are located on Union St, which has a decent amount of traffic (one lane in each direction). I highly prefer movers who have some experience with NYC moves like this--my understanding is that someone stands near truck, or you unload it all at once then move things in... it's a bit different than other places.