The distinctive meter of "Captain Kidd' and many other ballads is suggestive of some tunes, but not solid evidence for the identity of any one of them. There's till a lot of work to be done getting everything straightened out. The later tune or tunes are known as "Coming down/ Captain Kidd/ Admiral Benbow" and other titles. In the 1567 'Gude and Godlie Ballatis' is "My lufe is lyand sick, send him joy, send him joy" in the same meter, and "All my life, leif me not, leif me not" is close to it. C. M. Simpson in 'The British Broadside Ballad and Its Music' notes lack of solid evidence connecting tunes of various titles. 'The Diggers' Song' is the title of a short article by E. A. White in JEFDSS, IV, 1940, where the song (Gerard Winstanley's?) is given along with a long list of other songs with the same meter. White gives only the 'Pills' version of "Put in all" for a tune.

Simpson overlooked the earliest song, 1654, calling for the tune "Sound a charge, Sound a charge" ("Royal news, Royal news/ Touch and go, touch and go" were later titles for it). Simpson, with due regard that his identification may be incorrect, gives for these the tune "Put in all". (I'm a little surprised that I couldn't find this song and tune in DT.)

I've been looking for many years for tidbits of data that might clarify what titles can be solidly connected to what tunes, but I've only found two scraps that don't help much.

An earlier copy of the tune "Put in all" than that found by Simpson is in 'Twenty Four new Country Dances for the Year 1708'. This version is a bit different than the later copies. [There are 2 copies of this dance collection in Washington, DC, and none elsewhere. Title page is missing on the Library of Congress copy, and they have misdated it as 1714.]

In 'The Battle of Falkirk Garland Printed in the Year MDCCXLVI' is 'An Excellent new Song on the Jacobites and the Opression of the Rebels' 'To the Tune of, Captain Kid' [sic]. The 1st of 13 verses goes:

You Jacobites by Name, now give Ear, now give Ear,
You Jacobites by Name, now give Ear;
You Jacobites by Name, your praise I will proclaim,
Some says you are to blame for this Wear.

Subsequently verses have no relationship to those by Robert Burns.

There is an article by a folklorist on the versions of the tune "Captain Kidd", but I can't find my reference right now.

The apparent original broadside ballad of Captain Kidd is listed as ZN1837 in the index found in the Mudcat Forum links. That for the broadside on the executed chimney sweep "My name it is Jack Hall, chminey sweep, chimney sweep" from which the "'coming down, coming down" burden comes is unknown, but we have Cecil Sharp's traditional version (Songs from Somerset, iv, #86) I suspect the original tune for it was that called "The Sweep" that Thomas D'Urfey used for his song "When Soll to Thetis Pool, save the Queen, save the Queen" in 'Pills to Purge Melancholy', II, 1719.