In my very humble opinion, I believe that the movements and runs of any particular full-back are majority-wise the domain of the coaching staff and the individual assuming only the responsibility to carry out those instructions. For eg.
A full-back can provide attacking support in two ways :-
1. The winger remains wide and pulls the full-back wide with him; in that circumstance the full-back overlaps on the inside of the defender. This is quite common with City where Sane hugs the touchline and even someone like Delph, not a natural full-back, overlaps inside the full back to pull it back across goal.
2. The winger drops inside, pulling the full-back more central; in that circumstance the full-back overlaps on the outside of the defender, usually delivering a cross uncontested.
Both of the above are a direct result of attacking coaching and very specific tactical instructions. If the above doesn't happen, it's usually because of instructions for it not to happen for whatever reason it may be. No matter who we purchase, the main issue is tactical instruction. Even an ordinary individual can carry out the above instructions. It does not require world class talent to do so. Purchasing a better full-back won't automatically make this happen.