Question: Do we pick the element of the channeling execution for the entire logistics day (that is, not changeable until the next logistics period), or with each use of the channeling execution ability (that is, changeable with each killing blow)?
The Channeling Execution text: "Once this High Magic has been purchased, the character may Killing Blow using any one valid Channeling effect they have available by spending 5 Channeling points. This Killing Blow may be performed using any valid mechanism available (for example, a weapon they are wielding or their hands)." Emphasis is mine.
The effects you have available are dependent on your available sources. As long as you have a wand or relic of the flavor you which to killing blow with, you may use that element. In short, you can freely choose between different elements with a single purchase of Channeling Execution.