To be clear, Kemp's Jig has no relationship whatever with John Dowland and is not at all, to my ears or hands, in his style (being a lute player myself and a performer of John Dowland's compositions).

I don't know where the notion came from that the tune is associated with JD. I've only seen this claimed on the website, The Session. I have two theories.

1. 'This is a lute tune and John Dowland is the name of a lute player I've heard of (probably the only one), so it may be by him'. This is clutching at straws.

2. A version of tune, 'The Parlement', appears in the Folger Dowland MS. This manuscript includes compositions by Dowland in his own hand and with his signature, but this is not one of them. This tune is anonymous.

"I think Matthew Holmes transcribed lute tunes that he had heard so the tunes could have been around for some time before he wrote them down" is not necessarily accurate on either count. Holmes was an extremely able musician. While some of the pieces in his books may be his own arrangements this cannot be proven: they are as likely to have been copied from other contemporaneous collections. While it is true that music in the medieval and renaissance periods could have a considerable timespan of popularity, if the earliest evidence for a tune is, say, 1595, then we cannot claim an earlier date than 1595 without straying into the realm of claims without foundation.