If you paid 20M for your current plot and you use the relocation feature then you'll pay 13M for the Shirogane plot. The relocation is just an automated system to move your furniture from the current plot to the new plot and relinquish your current plot without having to do it all manually. There is no actual fee for the relocation beyond purchasing the new plot.

In the event you downgrade such as relocating from a Medium to a small, the 30% discount from the medium plot would be greater than the small plot so you would be able to purchase the small plot for free. But because the 7M discount is greater than the 4M purchase prices of the small plot, you will not be refunded the 3M difference.