I've already got the grow part down, where the particles will grow independently (using a particle texture). I need these particles to stick to my object while it moves. Using the "No" option for physics on particles will make them stick to the object, but only the Newtonion, Boids, and Fluid options will allow the particles to grow separately over time. But with any of those 3 options selected, the particles are then separated from the object, making them sit still and disconnect from the object as it moves. I need the particles to act like bubbles on the surface of the object. Haven't found a good solution so far.
Use Hair type instead of Emitter in Particle settings (suits all kinds of mesh deformations including armatures), it also provides object instancing.
Or stay with Newtone option, but use Lattice + Hook combo (for simple scale/rotate/locate animation).
Add Lattice object, scale it in object mode a bit bigger than your mesh Cube. Change interpolation to Linear.
Go into Lattice edit mode, select all vertices and press Ctrl H, select Hook to new object (Empty will be added)
Setting emitter object
Add Lattice modifier after Particle system and select your lattice object.
Now if you try to move Empty, it will engage movement for Lattice and also emitter as it is cotrolled by lattice object. So instead of animating emitter object, animate Empty.