Hmm.. that is definitely too early for immigration/border crossing records.
I would imagine they either moved looking for work or to be close to family? I'm not sure what the mining scene was like in Canada during the early 1800's?
OK... found a John Dungy living in Brant County, Ontario, in the 1861 Census. Occupation is given as 'grocer'; listed with Mrs. Dungy, no children. Baptist.
Found another John which looks like a match but in the 1891 Census living in Chatham Township, Bothwell County, Ontario (not terribly far from Brant County). John was 60 y/o (b abt 1831), born in the US, Methodist, widowed. Three others (John's children?) listed: James Dungy (1866), Joseph Dungy (1866), Mary Dungy (1877). All were born in Ontario, parents both born in the US. They are all listed as Baptist. Only John has an occupation (farm labour) and he is the only wage earner.
Not sure if these are related to your Dungy's or not. There are many more American born Dungy's as well throughout the Ontario records.