Taken from Census info...
They have children, Willard Miner and James when they leave New York, heading west...
1850, son James S. Buel, was born about February in New York.
In 1850, Isaac is living in Columbus, Columbia, Wisconsin, with wife Mary J. Buel, sons Minor W. and James A Buel. They appear, by the census found, to be living with William W. and Emily A. Buel, Isaac's older brother and sister in law. Possibly, they moved west together?
1852, daughter Francis is born in Wisconsin.
1860, Isaac is found 218 miles due west, in Mitchell Township, Mitchell, Iowa, with wife Mary, sons Miner Buel (14 years old), James Buel(10 years old), and daughter Francis Beal (8 years old)
On Feb. 16 1862, 42 year old Isaac Buell enlisted into the Grand Army of the Republic, Iowa 21st Regiment, A company, Inf., as Wagoner. He mustered June 11, 1862 and was discharged on April 15, 1863, at Cairo, Illinois.
During a search for Isaac, his son Willard Miner Buel was found in the same Iowa company. Per the census info from 2 years prior, Willard was 16, not 18.
Willard M. Buel, age 18, Residence in Mitchell, born in New York. Enlisted Feb. 16, 1862. Mustered June 11, 1862. Willard fought at Hartville, Port Gibson, Champions Hill, Black River Bridge, Seige of Vickburg where he was slightly wounded. He died at New Orleans, La. of small-pox Nov. 11, 1864.
Buried at;
https://www.findagrave.com/memorial/32431092/willard-miner-buell
In 1870, Isaac Buel(l) is in Iowa with wife Mary and son James & daughter Frances
In 1878, Isaac married at Osage, Mitchell, Iowa on 24 Mar 1878 to Betsy (nee Weber) Willey, born in Columbus, Pennsylvania, February 14th, 1832, died at Mitchell Iowa, February 13, 1907.
(Looking for death of wife, Mary nee Andrews, Buell)
In 1880, Isaac Buel, is living in Iowa with wife, Betsy and a boarder.
In 1900, 80 year old Isaac and wife Betsy, 68, are living alone at Mitchell, Iowa. Isaac dies 3 years later.
https://www.findagrave.com/memorial/22940723/isaac-buel
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If Issac was living with Mary in 1870 but by 1878 married his second wife then Mary must have passed away sometime between 1870-1878.