The first is July 4, 1866, when the flags were turned over to the state by the veterans. I need to dig into my files to find the references to the local newspapers that covered the story. However, in Volume I of Richard Sauers' "Advance the Colors," regarding the Pennsylvania flags, you will find on page 37, footnote 7, a reference to a story in the Philadelphia Press of July 4, 1866 with the headline, "Ninety Years (referring to the 90th Independence Day). Pennsylvania, Indiana, Michigan and Kansas To-Day Receive Back from their Soldiers the Battle-torn Standards."
The other is May 27, 1914, the day the G.A.R. Memorial Building in Topeka was dedicated. This building, which still stands across from the State House, was in part to be the home of the Kansas State Historical Society, which was being moved out of the State House. Surviving veterans carried the flags of their regiment from the display cases in the State House to the new display cases in the Memorial Building, a ceremony marred only by the fact that one veteran who was to carry a flag died of a heart attack that morning.
A quick source for this is the Autumn 1975 issue of the Kansas Historical Quarterly, particularly pages 339 and 342 (the issue deals with the 100th anniversary of KSHS. Again, the local newspapers provided good coverage of the story, and I'll try to get the exact citations up soon.
Meanwhile, here's a link to a photo of the veterans on the State House steps in 1914:
http://www.kansasmemory.org/item/211035