Paul, yes there are comprehensive musters for most of Cook's ship, on this website, go to
https://www.captaincooksociety.com/home/cook-s-ships/musters-of-cook-s-ships
But according to my records there was nobody called Dinsmore who served under Cook on any of his ships.

Is there a more complete crew list? I’ve been told there was a Dinsmore in one of the crews. Thanks

Joanna, thank you for your interesting enquiry. I wonder if the James Alexander who you are enquiring after was the same servant of the Earl of Saandwich as the James Alexander who was buried in the churchyard at Brampton? See page 19 of the following link
http://www.ely.anglican.org/parishes/brampton/pdfdocs/tBm1612w.pdf
That James Alexander came from Africa and not the Pacific Islands. To which I can add that details were recorded of several Polynesians who were passengers for part of Cook's voyages. But none were recorded as boarding at Fiji.

Is there any information on passengers picked up during the voyages?
I’m particularly interested to know if a man was taken to England from Fiji
The man was known as James Alexander when he was in the service of the Earl of Sandwich, supposedly brought back as a gift for the Earl to thank him for his support of the voyage

Nic, I have checked the master list of all the crew who served on Cook's three voyages of discovery, and the surname Peake does not appear. The musters of some of the other ships that Cook served on, including the Whitby colliers and some Royal Navy ships can be found on this website at
https://www.captaincooksociety.com/home/cook-s-ships/musters-of-cook-s-ships
have a look through those muster lists. There were 3 officers called James Cook serving in the Royal Navy in the late 18th century, I wonder if your ancestor might have served under a different James Cook
?

I've been told that one of my ancestors was a member of the crew on one of Cook's voyages - it may have been an earlier voyage than the 3 great ones. his name was W or T Peake - I was told he was a cook but he may have been a lesser rank or posting. Is there any way of finding out details of the other crew members? Id be grateful of any help Thanks Nic Peake in Manchester UK

Roger, yes Cook did have the services of a ship's clerk on all of his voyages.
The letter book of the Endeavour is held by the National Library of Australia, and is available on-line (I think). All of the letters in that book were written by Richard Orton, the ship's clerk, in a beautiful copperplate script, a far cry from Cook's handwriting. I expect that you know that the man in charge of the Marines on Endeavour was a Lt. Molesworth-Phillips .

Did Cook have a clerk (Midshipman) to write up his observation or did Cook do that himself?
I have a relative Richard Molesworth who was a Captains Clerk in the RN 1795-1809.

Eileen, thank you for your interesting enquiry. Yes, there was a William Charlton on the Resolution on Captain Cook's third voyage (1776-1780). But he enlisted on the ship in February 1776 as an AB seaman. Befire the end of the year he had been promoted to a Midshipman. At the time of his enlistment he was recorded as 18 years old, so his birth would have been about 1758, which ties in with your date.

I believe my ancestor WILLIAM CHARLTON signed on HMS RESOLUTION in Dec 1776 and i still there in Feb 1779, although his year of birth changes from year to year. He was actually born in 1758 at Essendon in Hertfordshire.