This refrain is rare, ocurring only in Child's examples F, H and O. The first two were from Motherwell's MS (p 383, Agnes Lyle, Kilbarchan, 27 July 1825; and p 147, I Goldie, March 1825) and the third from Peter Buchan's Ballads of the North of Scotland, II, 128: source indicated only as 'an old woman'.

Christie (Traditional Ballad Airs, 1876, I, 42; see also Bronson I, p 146, 10.3) prints a text 'epitomised' from Buchan's, with a tune he got from 'Jenny Meesic' of Buckie (Enzie, Banffshire). In a few places the text is amended from Christie's memory of her singing. 'Meesic' ('Music') wasn't her real surname, but a nickname inherited from her father, who seems to have acquired it towards the end of the 18th century by dint of local fame as a ballad singer.

Jenny provided Christie with 'a great number of old Airs and Ballads'; she died in 1866 at the age of nearly 80. It is her tune that Martin Carthy uses, a little modified in places. Pete and Chris Coe used a different tune, probably from another version of the song; though not from one with this refrain, as only the Christie tune survives for that. Both texts are re-written to varying degrees, in Martin's case with reference to Jodie Stecher's form of it.

In Child H, the line is actually given as 'Hey with the gay and the grandeur O', but the other two examples stick to 'grinding'. Not enough, really, to draw any conclusions as to whether or not there is (or, rather, was) any particular meaning to it. The large number of examples found in tradition are mostly of the 'Bow down' (England and America) or 'Binorie' (Scotland; though Bronson suspected that this was partly down to Walter Scott) varieties, though a number of other forms also turn up. The earliest example we know of, a broadside of 1656 (Child's example A, printed for Francis Grove of Snow-hill, London) has a fairly standard 'With a hie downe downe a downe-a'. Rimbault contributed this, with some commentary (including notes on the man who wrote it), to Notes and Queries 1st series, V (138), 591-2, in answer to an earlier enquiry in N&Q 1st series, V (127), 316-7.

Of course there are plenty of analogous songs to be found throughout Europe, especially in Scandinavia. An examination of refrains used in these might just perhaps offer some clues; though I rather doubt it.