Inspired by this question: Can you cast Mirror Image twice on successive turns and have 6 duplicates of yourself instead of 3?

Mirror Image (PHB, pg. 260) says:

Three illusory duplicates of yourself appear in your space. [...]

Each time a creature targets you with an attack during the spell's duration, roll a d20 to determine whether the attack instead targets one of your duplicates. [...]

A duplicate's AC equals 10 + your Dexterity modifier. In an attack hits a duplicate, the duplicate is destroyed. [...]

The PHB says in Chapter 10 under "Combining Magical Effects" (PHB pg. 205):

The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect - such as the highest bonus - from those castings applies while their durations overlap.

If you have two castings of Mirror Image on you, then you will only be under the effect of one of them at any given moment. However, which is considered the most potent as the one that is currently "active" starts losing duplicates?

For example, is it the case that the active one must lose all duplicates, then when you have none left, that spell ends and the second one "kicks in" and you have 3 again; or is it the case that as soon the first casting loses a duplicate, the second casting is considered active, so they effectively swap back and forth, so you end up with 3 duplicates, then 3, then 2, then 2, then 1, then 1, before finally having none when both spells have ended?

And would the answer change depending on whether one of them was cast with a higher spell slot than the other, since that's the other way I'd think to determine potency?

The spell with the most images would become active (all else being equal)

The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect - such as the highest bonus - from those castings applies while their durations overlap.

This is going to come down to how the DM rules how to measure potency as the game doesn't tell us a definitive answer for what should count. It seems reasonable to say that, all other things being equal, that the spell with the most images would be considered the most potent effect at the time.

How this would play out in practice would be that whenever a spell's number of images is reduced below that of the other spell, the other spell would become active. When both spells have equal images (and everything else) it really doesn't matter which spell is active done they are identical. The alternating behavior would continue until one spell loses all of its images.

What if all else is not equal?

The above assumes that the spells are identical in every respect (duration, AC, etc.) if there is any other discrepancies the DM will have to choose how they define potency in more detail. For example, if one mirror image was cast a round later than the other one it would be completely reasonable to rule that the one with the longer duration remaining would be the more potent spell and that it would override the spell for the entire duration. This would also be the case if one of the spells was cast at a higher level than the other (and everything else was the same).

The most potent casting or most recent would be in effect until it ends. Then the previous casting is in effect.

When the most recently cast mirror image runs out of images, it stops affecting the target. The previously cast instance of the spell is then in effect assuming it still has time on it's duration.

Casting the same spell multiple times

Relevant Sage Advice interview with Jeremy Crawford (~25:30) regarding person under the effect of a spell (true polymorph) and that spell being superseded by a subsequent casting of the same spell.

"The next one on the stack replaces the previous one on the stack unless the previous one ... is more powerful."

As pointed out by Rubiksmoose, the most recent errata for the Player's Handbook clarifies:

Combining Magical Effects (p. 205). In the first paragraph, the following sentence has been added to the first paragraph: “Or the most recent effect applies if the castings are equally potent and their durations overlap.”

They would switch out

The rule for combining Magical Effects simply says:

the most potent effect - such as the highest bonus - from those castings applies

It leaves determination of what for a particular spell counts as the Most potent effect to the DM & player.

For the Mirror Image spell that would in my opinion mean that the spell that gives you the highest chance of not being hit is used at any given time.

Equal potentcy

In this case when the bonuses are equal you have to determine which to apply. This is probably a call by the DM. Two options that I would call reasonable are:

When equal use the one that will end first.

For case 1 this would mean that after first casting spell 1 and then spell 2 then first spell 1 will be used. If spell 1 looses an image you use the 6+ roll of spell 2 then if that looses a image you would use spell 1 with an 8+ roll until it looses another image then you use spell 2's 8+ etc.

Switch spell only if it's better

For case 2 this would mean that after first casting spell 1 and then spell 2 then first spell 1 will be used. If spell 1 looses an image you use the 6+ roll of spell 2 then if that looses a image you would use spell 2 with an 8+ roll until it looses another image then you use spell 2's 1+ etc.

How it looks

In game I would explain the visual as you having 6 copies running about but that they keep moving through each other providing limited effect.