William Dun is listed as M.A of King's College Aberdeen in 1641 and said to have been tutor to Henry Duke of Gloucester the third son of Charles I, he took an M.D. at Leyden in 1653
[Bibliographia Aberdonensis 3rd Spalding Club MCMXXIX Vol. 2 page 363]

This petition failed to bring the desired result and he retired to France where he continued his practice - dates for him are unknown.

To the Kings most Excellent Majesty
The humble petition of Dr. William Dun, native of Aberdeen.
Now Physician Ordinary to the most Christian French King.
Sheweth,
That your Majesties Petitioner did serve your Royal father (of blessed memory)
In the quality of captain, under the command of the late marquis of Montrose, in
Scotland, And therefore exiled from there to Holland in the year 1649, where he was presented to your majesty as a loyal subject, at Breda, by the said Marquis of Montrose, Earl of Seaforth, and Major General Hurrie his kinsman: And there after commanded by General Ruthven then Chamberlain to wait (expecting the first vacant place) on your majesties Royal person, which he did accordingly through Holland, Flanders, and into France; where falling sick for a long time was separate ( to his great grief) from the service of your Majesty then going for Scotland, And after his recovery from a long and sad sickness in France, Did live by his practice in physic, which (by God's blessing) was so prosperous with him in the French Armies and Hospitals that made him at last famous, at the court of France, And promoted by Dr. Vallot's acquaintance (about the time of the French King's sickness at Calais) to be one of the King of France's Physicians Ordinary, in Chemistry ( as witness his Patent) a reward due to his desserts ; To the exercise of which charge he is to return to France again, how soon he shall be restored, by justice, to his patrimony in Scotland, whereof he has been deprived by the common calamities of the time these twelve years bygone.

Wherefore for his many sufferings & ... humbly desires

The honour and character only ( as a mark of your Majesties Royal favour to a loyal subject) to be sworn Physician Ordinary in Chemistry to your Majesty which will highly advance your petitioner and render him more capable to consecrate his ... and the rest of his life labours and talents to your sacred majesty. And for that and that you would be most graciously please to give order to the Lord Chamberlain that the paper be sworn And have his patent thereupon And he shall ever continue to pray

Background.

The English civil war had its roots in Scotland where Charles I attempted to impose English forms of worship in Scotland - this was resisted by the national Covenant of 1638 and developed into an armed struggle and it was to raise funds for this that Charles called parliament and Pandora's Box was opened.
Montrose was one of the first signatures of the Covenant - however as the Covenaters became more extreme and Montose felt that the King had conceded sufficient he took up arms against them.
In a brilliant campaign of 1644-45 winning six battles, Tippermuir,Aberdeen, Inverlochy, Auldearn and Kilsyth however he was caught unawares at Philiphaugh and his, by now depleted army, was destroyed.

Chancellor Hyde writing to Prince Rupert from The Hague where the court in exile was based under Charles II; (Charles I had been executed in London on 31 Jan 1649)

28 February 1649
Our court is full of Scots. The Earls of Lauderdale and Lanerick are here, being as they say, driven out of their country by the power of Argyll who is in firm league with Cromwell. Here is likewise the Marquis of Montrose, who is in truth a gallant person and very impatient to be doing; and though the Presbyterians are as busy as ever, yet I believe the next news I shall send you will be, that His majesty entirely trusts Montrose and puts the business of Scotland wholly into his conduct.
[Williams,R Montrose: Cavalier in Mourning, Glasgow n.d. page 327]

As ever Charles was surrounded by intrigue as this letter to Montrose from Elizabeth of Bohemia, (The Winter Queen) sister of Charles I, shows;

By great chance I have found that the Prince of Orange will again extremely press the King to grant the [Scots] Commissioner's desires and so ruin him through your sides. I give you warning of it that you may be provided to hinder it ... For God's sake leave not the King as long as he is at Breda, for without question there is nothing that will be omitted to ruin you and your friends and so the King at last. It is so late I can sy no more; only believe me ever your most constant, affectionate friend,
Elizabeth
[Williams,R Montrose: Cavalier in Mourning, Glasgow n.d. page 333]