What is the best way to algorithmically sort physical boxes?

Question

I work part time at UPS. Part of my job consists of taking boxes from in front of me, determining if the correct cage (1 of 6 moving cages) is behind me (true about 50% of the time), and then putting the box down either in the cage or back in front of me. There can be 6-20 boxes in front of me at any given time depending on box size (essentially the available storage for the sorting algorithm). Each time a box is removed it is replaced with another random box, if there is room for that box. If the next box is particularly large, 3-4 boxes may need to be removed first (meaning that removing a box does not guarantee a new box will appear unless the total number of boxes is sufficiently small). The number of boxes can be considered infinite (UPS does ship nearly all of Amazon’s boxes, after all).

I’m trying to figure out the best strategy for sorting these boxes, partially to work faster and partially for fun.

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A few more details

• 2 boxes cannot be picked up at the same time (they can be up to 70 lbs). The boxes can, however, be slid sideways. This means a box can be picked up and inserted between other boxes, but two non-adjacent boxes cannot be swapped in a single movement.
• Boxes can be stacked. However, the stack still needs to be put into the cages one by one. Stacking can reduce the number of comparisons needed (because you know all boxes in a given stack go to a certain cage), but not the number of movements required.
• Boxes must be picked up to be evaluated
• 1 cage will disappear and a new cage will appear every 10 seconds, in a predictable order
• If the correct cage is not behind me, I must either wait for the cage or set down the box on the shelf and select a new box

Computational time (seconds)

• Picking up a box: 1
• Evaluating a box: 2
• Placing a box in its original location: 0
• Placing a box in a new location: 1

Examples:

• Picking up a box, determining that the correct cage is not behind me, and putting the box back where it came from: 3 seconds
• Picking up a box, determining the correct cage is behind me, placing the box in the cage: 4 seconds
• Picking up a box from a presorted pile and placing it in the correct cage: 2 seconds
• Picking up a box and placing it in a cage without sorting: 2 seconds (plus the time to find a new job)

For context, the current rate is around 700 boxes per hour (5.14seconds/box). So reducing the average time per box by only 0.5 seconds is a fairly significant change.

What is the most time-efficient way to move the boxes from the shelf to the correct cage? Solutions to simplified versions of this problem are also welcome.