There is a "A History of the Edinburgh, or Queen's Regiment Light Infantry Militia (now) Third Battalion The Royal Scots" by Major R.C. Dudgeon. I have a copy and this seems to tally with what is on Wkipedia. My notes from these sources are -
In March 1800 the regiment went to to Ayr, with detachments at Newtown, Wallacetown and Stranraer; in June two companies went to Greenock with a detachment at Port Glasgow. Then in July the companies at Ayr moved into the barracks at Glasgow and took over guard mounting in the city.
The 10th North British was marched in January 1802 from Glasgow to quarters in Musselburgh, Fisherrow and Inveresk, and sent its battalion guns to Leith Fort. From these quarters the regiment could assemble at Dalkeith Park for exercises.
From 29 June 1803 the regiment was in camp at Musselburgh as part of a Scottish Brigade temporarily commanded by the Duke of Buccleuch. In November the regiment returned to Dalkeith, where temporary barracks were erected at nearby Lugton, with one company detached to Newbattle and one to Lasswade. It continued to take part in field days with the rest of the brigade through the winter. In 1804 a Scottish Militia Brigade was formed with its headquarters at Inveresk, and the 'flank companies' (grenadiers and light infantry) of the brigade were drilled together at Portobello, Edinburgh. The brigade also found the guards for a Prisoner-of-war camp at Greenlaw House (later Glencorse Barracks). In May 1807 the Edinburgh Militia moved from Dalkeith to Dunbar barracks, with one company detached at Eyemouth barracks. The duty was to assist in preserving the peace and aiding the revenue officers. It returned to Musselburgh in February 1808, then went back to Dunbar in February 1809. In April 1809 it went to Haddington barracks , and then to Edinburgh Castle in June, with a detachment at Leith Fort. The duties included escorting parties of prisoners of war.[29]
In March 1811 the Duke of Buccleuch resigned command of the regiment and was succeeded by the Earl of Ancrum (later 6th Marquess of Lothian)
As in the July of 1811 the regiment left Scotland for the first time, sailing from Leith to Harwich where it went into barracks before being stationed at Colchester for nine months. In April 1812 the regiment moved to the large prisoner of war camp at Norman Cross Prison. Following an outbreak of Luddite machine-breaking in the English industrial districts, in May the regiment was marched to Nottinghamshire, being quartered at Mansfield. In the October it was quartered at Manchester with detachments at Bolton, Bury and Rochdale, and after short stays in Birmingham and Somerset in February and March 1813, the regiment marched to Plymouth, being quartered in Frankfort Barracks from 31 March.
Most of the Edinburgh Militia sailed from Plymouth on 26 June 1813 and disembarked at Monkstown, Ireland on 12 July. It was then marched towards Armagh, but diverted to Derry where it arrived on 31 July.
The war having ended with the deposition and exile of Napoleon in 1814, the regiment was relieved in February 1815 and marched to Belfast where it embarked for Scotland. It reached Stranraer on 9 March and marched to Dalkeith, where the men were given leave before the regiment was disembodied on 3 April 1815. The Regimental Colours were laid up in Newbattle Abbey, the Marquess of Lothian's house. Unlike some regiments, the Edinburgh Militia was not re-embodied during the short Waterloo campaign of 1815.
Hope this is of help