Well there are two modes of convection which operate in the Earth's mantle (and yes the mantle convects and that is well established): 1. Mantle plumes, 2. Broad scale convection typical of a fluid heated from within. There is actually some debate (although it is becoming increasingly muted) as to the existence of mantle plumes (the evidence seems to suggest that there are), but there is no debate as to the existence of convection in the mantle, in fact convection is how the Earth transports over 99% of its heat near to the surface, and this is widely accepted.
Now to tackle another apparent misconception. Mid ocean-ridges are where mantle material is upwelled, but contrary to your understanding, mid-ocean ridges are pulled apart and material flows up to fill the gap. They are completely passive features. It is not that forced upwellings push the plates apart. Rather, to the contrary, as structural geologists have observed, mid-ocean ridges are being pulled apart rather than being pushed apart, and this is why there are extensional faults in Iceland for example. Mid-ocean ridges form wherever they happen to be, so yes, they can move, but not because the convection cells move.
As to the nature of convection cells in the mantle. I think that nobody really knows for sure what they look like, but they almost certainly aren't the neat cells pictured in certain schematic cartoons. There are certainly large downwelling areas related to the subduction of ocean lithosphere at the subduction zones. These subduction zones tend to occur where continental plate meets oceanic plate (with some exceptions) and we know that continents can move, and indeed that they can come together to form super continents (e.g. Pangea). The coming together of super continents requires that everything in between the super continents must be subducted to make room for the continents to come together. So we can envisage a massive downwelling beneath the super continents. But then when the continents have come together there is nothing left to subduct, apart from a bit of dense continental root, perhaps. And so the downwelling drys up. Eventually there must be upwelling beneath the continents which forces the continents to break apart. So it seems that mantle convection must move (or perhaps completely change their character) in order to recreate the supercontinent cycle. So I guess the short answer to the question would be: yes.