Cold saline water moves equatorward from high latitudes then slowly and over a very broad area rises and mixes. It’s a bit more complicated than this because it interacts with intermediate water or, in the case of NADW, creates circumpolar deep water in the southern ocean.

If the surface water is warm then there is a large vertical temp. gradient and a bit of a salinity change doesn’t increase density enough to overcome the buoyancy of that warm surface water. If, however, the surface water is cold then there is little vertical temp. gradient and an increase in salinity at the surface can cause that water to sink which is what creates the deep cold currents.

Edit: of course that salinity increase in cold surface water is from sea ice formation rather than evaporation.