In 5e, silence is cast on a point you choose within range. There is no mention of it moving, or being cast on an object or creature. It creates a zone that does not move.
An extremely liberal interpretation of the spell could allow that the point chosen is an object and that it can subsequently be moved, but this is contrary to rules-as-intended. Note that pretty much every other "area effect" spell uses exactly the same terminology. You cast a fireball at a point you choose. You cast an earthquake on a point on the ground. Nobody is going to allow that you can cast an earthquake on a stone on the ground, and then pick up the stone and move the area of effect (...although, having said that, that'd be pretty cool). The same is true for cloud of daggers, confusion, and many other spells. All use the phrasing "at a point you choose within range". If they can be cast on a single creature or object and then move with that object, it'd create a balance nightmare.
So, no. In 5e, a silence spell does not move.
