The swan maiden tale keeps a remarkably stable shape across an enormous geographic range, which is exactly what earns it a numbered entry of its own in Stith Thompson's Motif-Index of Folk-Literature rather than a loose family resemblance noted in passing. A man, often a hunter or a young man out alone, comes upon a group of swans that shed their feathered skins by a body of water and become women; he steals or hides one skin, preventing that swan-woman from returning to her flock and her bird form, and she is bound, in most tellings against her will, to stay with him, frequently as his wife. The marriage lasts for a fixed or unstated period, sometimes producing children, until the swan-wife recovers her hidden skin, whether by finding it herself, being told where it is, or being given it back, at which point she puts it on and departs, generally without looking back and often permanently.
What the Motif-Index makes visible by cataloguing so many regional versions side by side is that the tale is not really about swans specifically so much as about a supernatural bride whose consent to the marriage was never secured and whose autonomy is restored the instant her stolen object is returned; the same underlying structure recurs with seal-skins, feather-cloaks of other birds, and other magical garments standing in for the swan-skin depending on the region telling it. The swan version is simply the best attested and most widely distributed single variant, appearing independently in Old Norse, Irish, Russian, Central Asian and East Asian material without an obvious single point of origin any scholar has been able to establish.