Well, since you are interested in the detailed movements of Co. K of the 1st LA. Cav. (Union) only, my suggestion would be to check the entire listing of the men of that company's individual service records. This objective can be completed by either purchasing the film directly from the National Archives or by finding an institution that has the film and using their copy for research.
Unfortunately, for your particularly detailed reasearch, the film is alphabetical by the entire unit, not by company. Companies sometimes have a listing of places they were at in this film besides the normal service records.
When looking for where a company was stationed, pay close attention to the documentation of Company K's officer's requisitions and where they were written from and the dates. It should provide some clues. The requisitions were not always included but a lot were so you may get lucky.