Why is the Hen joined with the Crane?

Not hard. when the harvest is carted, and the gleaners have gone, the Hen is turned into the cornfields to fatten on what she can find. And a mouse-coloured little rival creeps around with her.

The White Goddess, Robert Graves, pg. 298

Black Hen

In the story of Gwion as he is being persued, his transformation occurs in seasonal order; the hare in the autumn coursing season, the fish in the rains of winter, a bird in the spring when the migrants return, and finally as a grain of corn in the summer harvest season. His persuant rushes after him first in the form of a greyhound bitch, then of an bitch-otter, then of a falcon, and finally overtakes him in the shape of a high-crested black hen - the image of the Death Goddess, representing the end of the solar year in the winnowing season of early autumn.

White Goddess, Robert Graves, pg. 400