So, I had the exact scenario last night, but at a less critical, less public facing area of my network.
Once both layer 3 switches are in place, with the new one connected to a trunk port on the original, all interfaces and static routes (logic) moved to the new switch without issue. I did try to just move the interfaces at first, but Meraki Dashboard forced me to move all routing logic to the new switch, not piece by piece as I intended. There was no service interruption, however.
After I was comfortable with the routing, I replaced the original layer 3 by moving it's uplink to the new layer 3. This re-patch had a very short outage, as expected.