Can the intangible illusory copies obtained by casting mislead move through walls?
If the illusory copies cannot move through walls, what would happen if an illusory copy tries to push someone else?
From the rules text:
the invisibility ends if you attack or cast a spell (PHB. p. 260)
I am making an interpretation here, because the character controls what the illusion does, and "you attack" seems to fit with "the illusion attacks" depending upon whether you are moving your illusory double, or if you have switched back to your own senses. It's unclear if the illusion can attempt an attack at all. A push fits into the category of shove which is a special form of melee attack. (p. 195, PHB, Shoving a Creature).
Using an attack action, you can make a special melee attack to shove a creature ...
Moving through walls is not addressed in the rules, so there you have to get a ruling based on either "if it doesn't say I can, I can't" or "if it doesn't say I can't, I can" which is as much a DM/table philosophy as anything else.
Since the general intent in D&D 5e is that "spells do what they say they do" it's a fair ruling to say "No, the illusion created by mislead doesn't move through walls."
Moving through walls, if allowed, would likely give away the illusion to any other observer, so as DM it would be appropriate for anyone seeing the mislead-created illusion moving through walls to have advantage on an ability check to notice that it is an illusion, or to recognize it as an illusion of your character without a roll.
(The comparison to Project Image has been removed now that the question has been updated).