Strictly RAW: No your illusion can't move on it's own.
Phantasmal Force (emphasis mine):
Choose a creature that you can see. It must make an Intelligence save. If it fails, you create a phantasmal phenomenon of your choice that is no larger than a 10-foot cube. It's only visible to the target.
While a target is affected by the spell, the target treats the phantasm as if it were real. The target rationalizes any illogical outcomes from interacting with the phantasm. An affected target is so convinced of the phantasm's reality that it can even take damage from the illusion. A phantasm created to appear as a creature can attack the target. Each round on your turn, the phantasm can deal 1d6 psychic damage to the target if it is in the area of or within 5 feet of the phantasm, provided that the illusion is of something that could logically deal damage.
Jeremy Crawford's tweet with the bag is RAW because of the emphasis part above.
The reason I say it can't move by itself is what's missing in the description when compared to Silent Image:
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.
"Spells only do what they say they do."
And there is nothing about your illusion moving in the description of the spell.
I've changed my answer from Yes to a No, this is the old answer:
Your phantasmal phenomenon can move freely. There is no limitation on the movement of your illusion. Only limitation is that it can't be larger than 10ft cube, not that it must remain within a 10ft cube. Your phantasmal phenomenon can be a little killer bunny that chases and attacks your target relentlessly for example, even if your target leaves the range of the spell afterwards the bunny can keep chasing because the spell doesn't say it's broken if your target leaves spell range.
Justification of this is that 'phantasmal phenomenon' is not specified, you can create the illusion of whatever you want. The only limitation of the illusion is that it must fit within a 10ft cube and if able to deal damage it has a 5ft reach to do so. You can say your phantasmal phenomenon is a hit&runner owl that pecks at his eyes or a clown that runs around mocks your target. Nothing in the spell description says your phantasmal phenomenon has to be static.