Not really, but...
This is a brilliant question and it reminded me of the versatility of illusion magic. It also made me reflect on what is natural movement and a passively animated illusion.
Silent Image requires agency on the spellcaster's behalf by using their actions to concentrate on the movement; it is a condition of the spell that needs to be met for his to occur. This is very different from Programmed Illusion for instance where, once the illusion is activated, the "scripted performance" will continue for up to five minutes (PHB p.269).
You can use your action to cause the image to move to any spot
within range. As the image changes location, you can alter its
appearance so that its movements appear natural for the image. For
example, if you create an image of a creature and move it, you can
alter the image so that it appears to be walking. (PHB p.276)
In Major Image this is also a requirement so you would need to still use your action to create the illusion of natural movement from one spot to another.
As long as you are within range of the illusion, you can use your
action to cause the image to move to any other spot within range. As
the image changes location, you can alter its appearance so that its
movements appear natural for the image. (PHB p.258)
Interestingly, in the first version of Major Image for 5e there was a secondary option when cast at a Higher Level, which allowed you to create a five-minute illusion that does not require concentration or for you to use your actions to make the illusion move naturally. In the spell it was described as a Permanent Image. I suspect that this was present to provide a Sorcerer access to this feature, as this class does not have access to the 6th-level spell Programmed Illusion.
At Higher Levels. When you cast this spell using a spell slot of 6th
level or higher, the spell lasts until dispelled, without requiring
your concentration. In this form it is sometimes considered a
different spell, known as Permanent Image. (Basic Rules p. 96)
There seems to be a pattern here, because both Programmed Illusion and the old Permanent Image version of Major Image need to be cast as a 6th-level spell. I think this gives a clue as to the level a spell needs to be for an extended natural-looking illusion to take place without the caster needing to expend their actions to control the image and make it move naturally.
If it doesn't work for these higher spells I think it is fair to say that it will not work for Minor Illusion cantrip.
Maybe... the clue is in the Campfire
The condition set out for something moving from one spot to another is important in this question. In previous answers it suggests that you could create a "passively animated illusion" of something like a campfire but this must remain on the same spot.
More about the campfire here:
Is Silent Image animated if an action is not used to move it?
In the end it will be up to the DM to determine these limits.
Personally, I would allow the creation of something like the illusion of a Bonsai tree on a windowsill that appears to be swaying in a light breeze. Or, steam rising from a freshly backed apple pie. Maybe, even a rabbit that stays in place eating grass or a blackbird that stays in the same spot, pecking at some bird seed in a repetitive way.
I think anything more elaborate than this which is not repetitive would require the caster to either use their actions to manipulate the illusion realistically or you would need to use a 6th-level spell slot.
As a DM, I would not allow for instance the image of a humanoid sitting at a table and having a meal or making a cup of tea. This would require the caster to use their actions to control the illusion or the use of a 6th-level spell.
At a push, I might allow an image of a humanoid sitting placidly, staring vacantly into space, where there is very minor responses such as blinking or his/her chest rises and falls with their breath.
I hope this is helpful.
