As Michael Bowen was the first to point out, increasing size is the key to this riddle about hawk names. His suggestion–Crane–is a good one, giving us Sparrow (Hawk), Pigeon (Hawk), Duck (Hawk), Crane (Hawk).
Andy Jones kept it neatly within the family Falconidae with his proposal, Partridge (Hawk), an obsolete name for (among other species) Gyrfalcon.

Photo: James Lidster
Both answers count as correct, while Elwood Hain’s answer, Auk, wins the prize for the most compelling and most poignant series submitted: (Dusky Seaside) Sparrow, (Passenger) Pigeon, (Labrador) Duck, (Great) Auk.
My own answer? Goose. Falconers are said to have classed their birds by size, the smallest being the Sparrow Hawk, followed by Pigeon Hawk (Merlin) and Duck Hawk (Peregrine Falcon), all of them outweighed by the Goose Hawk–which we know as Goshawk.
Michael, Andy, and Elwood will all soon be sporting their stylish new WINGS caps.