Mushet was born in 1716 at Dublin of a Jacobite family, who had fled there from Stirling.
He is supposed to have been educated at Trinity College, Dublin, and was entered at Leyden on 26 August 1745 (Peacock, Index, p. 72).
He was made physician in chief to the forces, and served at the battle of Minden (1759), but declined an offer of a baronetcy for his services in that campaign.
Mushet was intimately connected with the Duke of Rutland, and had apartments for eleven years at Belvoir Castle.
A monument was erected to his memory by his daughter Mary in the church of St. Mary Castlegate, York, with a long inscription written by Sir Robert Sinclair, recorder of York.