In such cases, nothing actually changes infinitely quickly, save the point of view of an observer of the field. For example, when an observer begins to move with respect to a static field that already extends over light years, it appears as though "immediately" the entire field, along with its source, has begun moving at the speed of the observer. This, of course, includes the extended parts of the field. However, this "change" in the apparent behavior of the field source, along with its distant field, does not represent any sort of propagation that is faster than light.