Hi, Jpr. I Googled for "wichita march" music and came up with an interesting entry from the Collections of the Kansas State Historical Society, Volume 12 (click).

So, Captain and Brevet Major Earl Van Dorn was born at Port Gibson, Mississippi, on May 8, 1820. He graduated from West Point in 1842. In 1858, he was stationed at a permanent camp in the Wichita Mountain Range. On October 1, 1858, the cavalry attacked a band of Comanches near the Wichita village adjacent to the present site of the town of Rush Springs, Oklahoma. The Comanches were taken by surprise and were dispersed, with all of their horses and fifty of their people killed. A U.S. officer and three men were killed, and Major Van Dorn's body was penetrated by an arrow. Van Dorn went home to Mississippi to convalesce.

There he was feasted and feted and honored by admiring fellow citizens, while one of his sisters, who was a gifted musician, composed and dedicated to him a piece of martial music, entitled 'The Wichita March,' which straightaway became a great favorite with the regimental band of the old Second cavalry.

So, this "Wichita March" was written by Major Van Dorn's sister, apparently in Port Gibson, Mississippi. Of course, it might not be the same march as the one that interests you. My thinking is that most marches don't have lyrics...but maybe the one you seek does.

When I visited there in the 1980s, Port Gibson was a sleepy little town on the Mississippi River. It had a nuclear power plant under construction at the time, so I suppose it's not so sleepy now.

Rush Springs, Oklahoma, promotes itself as The Watermelon Capital of the World." It's southwest of Oklahoma City, about halfway to Wichita Falls - so it's not at all near to the City of Wichita in Kansas.