It can be done with chalk, but it is the most difficult, in my opinion, to do a good job that way. The problem is that, unless one uses the Bragdon-type powders that also have an adhesive property, they'll rub or blow off. What we do to prevent this is use a matte medium fixative, or something like Dull-Coat from Testors as an covering. But when you use that covering, the chalk you built up all but disappears, so you have to make multiple coats of powder and Dull Coat until you achieve what you are looking for.
There is a lot to be said for an airbrush, but it's only as good as the hours of practise and talent permit. And, not all airbrushes are created equal. Different designs have different advantages.
I have gotten good results with using heavy washes of blends of cheapo acrylic crafts paints. I wipe most of the wash off on a paper towel, and then swipe the item to be dirtied. Again, it takes practise, and should be done on junk first. That allows you to warm up and to get the right colours before you actually embark on any one weathering session.
Here is a PCM Norfolk & Western Y6b that I recently weathered. It isn't close in the shot, but close enough to see the overall effect. The coal hoppers and caboose were also done with washes and a dryish brush method.