There's an interesting document, "The Union Defenders of Kansas," which includes a sort of census, or survey, of Union veterans residing in Kansas towards the end of 1865. Several Arkansas veterans are mentioned living in Kansas with their families. I tracked a representative sample through the 1870 U.S. census of Arkansas, and found that they had all moved back to Arkansas. They had apparently fled the state at some point during the war and, once they had established their families in safety in Kansas, joined the Union army. Once things had settled down, they returned to their former homes.

On the other side of the coin, a fairly large number of people fled Arkansas and moved to Texas -- lots of Crawford County families for some reason -- and, after settling their families, enlisted in various Texas regiments. Most of those refugees remained in Texas after the war.