There is no evidence that the Dundas Militia carried regimental colours during the War of 1812. General Prevost requested flags for Canadian militia regiments but most did not arrive until years after the war's conclusion.
However, there is evidence that the 3rd battalion York militia carried King's and a Regimental colours, which were consecrated on 23rd March, 1813. The remnants of these colours still exist.
Also, several Lower Canada regiments had regimental pairs of colours. Further, the militia regiments of New Brunswick and Nova Scotia carried regimental colours. Ref: "Scarlet Coat: Uniforms, Flags and Equipment of the British in the War of 1812", Rene Chartrand, Service Publications, 2011.pp 181-184.
Again, there is no evidence that the Dundas militia regiment possessed regimental colours but if they did it is likely they would be similar to the British regular's flags.