This is something called the metric expansion of space. The metric value is changing at a local level where the measurement is taken. I think it is misleading to talk about space as a dimension moving at some speed, because speed is really distance per unit time. the expansion of space is just space per space changing. We tend to embed our explanations as if the universe expanded from a point on a piece of paper, it doesn't really work that way, the paper in a sense is being generated as it goes along. The speed of light is just some measured velocity value that happens to be pretty close to c when we measure it at various places around the universe, light goes through the metric of space. The metric of space doesn't really say much about what that speed is. So c just goes along the metric as the metric is generated.