Sources of information for Carrie and David Nation which are on the Barber County, Kansas: History and Geneaology site are as follows:
Biography of Carry A. Nation:http://www.rootsweb.com/~ksbarber/nation_carry.html
Obituary: Captain David A. Nation:http://www.rootsweb.com/~ksbarber/nation_d.html
The Use and Need of the Life of Carry A. Nation, by Carry A. Nation, Revised edition, 1905: http://www.rootsweb.com/~ksbarber/nation_carry_useandneedboo...
or alternate url:http://tinyurl.com/bwxvn
The Rootsweb Message board for the surname Nation is here:http://tinyurl.com/r2kjm
So far as I've found in my reading, Carrie Nation had only one child, by Dr. Charles Gloyd; they were married on November 21, 1867. Gloyd was, by all accounts, a severe alcoholic; they separated shortly before the birth of their daughter, Charlien.
According to the obituaryhttp://www.rootsweb.com/~ksbarber/nation_d.html
for her 2nd husband, Capt. David A. Nation, she had no children by him.
I googled for info. on her child and have found, thus far, the following:http://www.tsha.utexas.edu/handbook/online/articles/NN/fna7....
"Belton, Missouri. There on November 21, 1867, Carry married Dr. Charles Gloyd. They became the parents of a daughter, but Gloyd's alcoholism had destroyed the marriage even before the child's birth, and he died shortly after Carry's parents forced her to return home. She then enrolled in Warrensburg Normal Institute and, after receiving a teaching certificate, taught school in Holden, Missouri, for four years. In 1877 she married David Nation, a lawyer, newspaperman, and sometime minister in the Christian Church."
It appears that you must descend from her daughter Charlien Gloyd if you are Carrie's descendant. However, I do not know if Charlien had any descendants.
David Nation, however, did have children from his first marriage who were mentioned in his obituary.
I have no connection with the Nation family except for the fact that my paternal aunt, Helen A. (Ferrin) Flory, and her husband, Charles Wayne Flory, were either the last private owners or among the last private owners of the Carry Nation house in Medicine Lodge, Kansas, in the early 1950's.