Hi Illusion_rox,
Let's see if we can get you an answer. Let's make sure I understand your question first though. Paraphrased, you are wondering how (multi-site?) vxlan fabrics performs routing decisions across the data-center interconnect (DCI).
VxLAN did find it's roots in what's called Flood-and-Learn VxLAN. In this type of fabric, end-device location is learned through the data-plan (flooding the network) with no control-plane help. There are many limitations to this type of design.
Modern VxLAN fabrics almost always implement a control plane of some sort. COOP for ACI, LISP for SDA, and BGP EVPN for Nexus standalone to list a few.
BGP-EVPN VxLAN fabrics utilize BGP for overlay decision making (end-host tracking is found in the overlay). The actual physical path (or hops) taken to get from host A to host B now becomes an underlay (IGP) problem with the overlay (BGP-EVPN) providing only the IP to head towards; whether it is a Border-Gateway or the switch host B is hanging off of.
This separation of the overlay and underlay is a huge advantage to design and tremendously simplifies the answer to your question. You can treat your uplink as a normal routed interface. For instance, you can alter link preference by manipulating the costs or weights (protocol depending) of your uplinks to influence how traffic leaves your network.
If your interested in learning more about vxlan multi-site design, you can that here:
https://www.cisco.com/c/en/us/products/collateral/switches/nexus-9000-series-switches/white-paper-c11-739942.html
