Yes, you can move the image out of the original range

"Within range of the illusion" refers to the 20 ft cube illusionary object or creature, NOT the 120 ft casting radius.

As long as you are within 120 ft radius of the illusionary object, you may move it to any other point within range. This lets illusionists keep one persistent Major Image for any amount of time if they can keep using actions to keep it near them.

What defines "any other point within range" however, is up to discussion. It could mean you may move the object either 120 ft within the caster, or 120 ft within the illusionary object.

Let's consider this from two perspectives: RAW and RAF.

RAW: The Mirror Image duplicates are only around the original target of Mirror Image (Illusory you, or real you)

The only official guidance we have here may be seen to confuse more than clarify. It is found in a tweet from Jeremy Crawford:

Question: If you have mirror images out and then cast invisibility on yourself, do the mirror images also become invisible?

Jeremy Crawford: Mirror image creates illusory duplicates of you that imitate your appearance. If your appearance goes away—you become invisible, for example—then the duplicates imitate that too.

This seems to give us clear guidance at first: a spell cast upon you also seems to apply to your illusory duplicates. But note that the spell in question (Invisibility) altered your appearance: but the illusory duplicates from Mirror Image do not change your appearance, but rather create the appearance of other things around you.

Invoke Duplicity creates a "perfect illusion of yourself" (PHB, p. 63). It does not necessarily create an illusion of the effect of spells that are near you. As an example, if you had the spell Minor Illusion to create the illusion of a 5' cube of granite in your own space, then used the Invoke Duplicity ability (or did these two things in the opposite order), your Duplicity illusion would not also have a block of granite surrounding it, because the granite is not part of "yourself." Similarly, the Mirror Image duplicates flitting around you are not "yourself," (although they look like you), and are thus not automatically recreated by the Invoke Duplicity illusion.

However, you could certainly cast Mirror Image through the illusory double (Case 2), and give it (but not you) several Mirror Image duplicates, since, as you mentioned:

you can cast Spells as though you were in the illusion’s space (PHB, Trickery Domain)

and

Three illusory duplicates of yourself [from Mirror Image] appear in your space. (PHB, p. 260)

Even though Mirror Image has a range of "self," the feature that permits you to cast Spells as though you were in the illusion's space is designed to expand the range of your spells so that it appears that the illusion is the one casting them. For example, a multiclass character could cast Thunderwave (range of "Self (15 foot cube)") centered on the Invoke Duplicity illusory version of themselves. Also, note that it has been established that you could cast spells with the range of touch against targets that are adjacent to your Invoke Duplicity illusion. Similarly, you could cast Mirror Image centered upon your Invoke Duplicity duplicate, rather than upon yourself.

RAW, it's debatable whether they would then proceed to mimic your actions or the actions of the illusory duplicate, but regardless, they would certainly appear in its space.

RAF: Sure, duplicate the Mirror Image doubles too. Who cares? And it's cool!

The RAW reading above is reasonable, but inherently problematic. If the Mirror Image duplicates don't count as part of your appearance, do your clothes? Does your Light cantrip cast upon your shield? What exactly does count as "you" for the purposes of this illusion? The argument leads us down a slippery slope.

Also, giving the Invoke Duplicity illusory version of you access to the Mirror Image duplicates that flit around the real you for free when you cast Mirror Image will have no mechanical effect on the game. Note that, unlike the Mirror Image duplicates of yourself, your Invoke Duplicity illusion does not disappear if it is struck, but only when 1 minute passes (or you drop concentration), so there's no benefit to making it harder to hit. Also, unlike the Silent Image illusion spell, it does not become faint or ineffective if someone realizes that it is just an illusion. It will still permit you to cast spells as though you were in its space and give you advantage on attacks against creatures that both you and it are within 5' of even if everyone in combat realizes which of you is real and which is an illusion. The mechanical effects of the Invoke Duplicity Illusion do not at all depend on the opponent knowing (or not knowing) which one of you is real.

So giving the Invoke Duplicity illusion three more illusory versions of itself (that remain in its space) whenever you cast Mirror Image on yourself (or if you already cast Mirror Image before creating the Invoke Duplicity illusion) would have little to no mechanical effect on the game, other than to make the combat more chaotic. This is especially noteworthy because this is an effect that a player could already create RAW by casting Mirror Image twice, once on themselves and once on their Invoke Duplicity illusion (since it is not concentration, and will validly apply to two different targets without invoking the "combining magical effects" rule). Requiring a player to cast Mirror Image twice seems like an unreasonable waste of resources for something with no mechanical benefit, so as a DM I would permit allow the Invoke Duplicity illusion to have the Mirror Image duplicates around itself in both Case 1 and Case 3.