Had a chance to check the census record today and here's what I've got:

1850 Census, Frankfort Twshp, Will County, p. 253
17, Lemuel Churchill, 41, Farmer, 200, VT
Sarah, 39, VT
Orson, 19, Farmer, VT
Lucy, 13, VT
Mary, 9, VT
John, 4, VT
Sarah J., 1, IL

1860 Census, Mokena, Frankfort Twsp, Will County
Saml Churchill, 52, Farmer, 2000, 700, Vermont
Sally, 49, Vermont
Orvis, 25, Laborer, Vermont
Mary, 19, VT
Jane, 11, IL
Samuel, 5, IL

1870 Census, Frankfort, Will County, Pg. 2, 5-Jul-1870
Churchill, Lemuel, 62, Farmer, 4500, 982, VT
Sally, 59, Keeping house, VT
Jane S., 21, At Home, IL
Lemuel, 14, At Home, IL

After this, the head of household is Lemuel, Jr.:

1900 Soundex, Will County, Frankfort Twshp, 158, 105, 3, 33
Churchill, Lemuel, July 1855, IL
Alida, Jun 1866, NY
Albert, s, Mar 1886, Iowa
Alice, d, feb 1888, Iowa
Sarah, d, Jan 1890, Iowa
Grace, d, Oct 1891, Iowa
Mary, d, Dec 1893, IL
Susie, d, Jan 1898, IL


1910 Census Frankfort, next door to Charles and Jane Fern
Churchill, Lemuel, 55, IL, VT, VT, Farmer
Alida, 47, Iowa, NY, unk.
Albert, 20, Iowa, IL, Iowa
Grace, 18, IL, IL, Iowa
Mary, 16
Susie, 11,IL
Ernie, 9, IL
Margaret,3,IL


1920 Soundex, Will County, IL 208, 154, 7, 70
Churchill, Lemuel, 64, IL
Elida E., 53, NY
Albert J., 33, Iowa
Earnest E., 19, IL
Robert H., 13, IL
Irvin L., 9, IL


It appears from this that Lemuel, Sr. moved from Vermont to Illinois around 1845/46 and that Lemuel, Jr. (at least) moved to Iowa sometime in the 1870s or early 1880s before returning to Illinois in 1892/93. I found no Lemuel Churchills in the 1880 Illinois Soundex and ran out of time before I could check the Iowa one.

Oddly, there seems to be another Lemuel Churchill floating about, as there is a marriage record in Cook County for a Lemuel Churchill marrying a Sophie Hewitt on 2/25/1881.
See:
http://www.cyberdriveillinois.com/cgi-bin/archives/marriage....

Also saw a few other Churchill marriages in Illinois involving Lemuel Sr.'s other children. For instance, Orson Churchill married in 1861, died in Nashville during the Civil War, then you see his widow Esther remarrying in 1866.

The next step would be to pull some of the vital records. I've dealt with the folks in Joliet and they have been quite helpful with the Will County records, however, when it comes to Cook County, I prefer to deal with them by mail, as there tend to be long lines and parking is a hassle in downtown Chicago. I live in the Northeast, so if you could help out with some of that legwork, I can probably spare a day or two to check things out in Vermont.