JacobRyan, I think I've found a solution to the problem of Hazel trying to move files outside of the specified time. Rather than setting a specific time such as 2:30, I set two rules. In the first rule, I tell it to only run after 2:30 and the second says to only run before 2:31. The seems to prevent any subsequent files from being moved over automatically. Of course, this assumes you will not be adding any additional files in that 1 minute span of time. In my case, I am running these rules in the middle of the night so no other files should be added to the folder. Here's what my recipe looks like now:

Mr_Noodle, I now have the rule set to run when conditions are met for "the file or folder being matched" and it seems to be moving the files I have targeted, which takes care of my original issue. The new problem is that if I set the time to run the way I did in my original screenshot, it will move the targeted files at the specified time, but it will also move any files I add to that folder after the specified time. The solution I mention to JacobRyan in this post seems to work for me, but it does not seem to work properly when I make a condition that Current Time IS a specific time. Any suggestions?