Lets say I have two spheres of equal dimensions, one charged and one uncharged. Now I connect them with a conducting wire. They will now very quickly reach equal potential. Can it be said that the total charge on each sphere remains almost unchanged?
You haven't specified if the spheres are conducting or dielectric. If it's the former, charges will flow from the first sphere to the second until they are of equal potential. How much charge flows will depend on their capacitances. Actually the final amount of charge will be proportional to their capacitance. However, if the spheres are dielctric, only the surface charge of the first sphere will be shared between the two spheres, because the volume charge of the first sphere cannot flow to the surface.