If you mean:
"will a LoRaWAN node connected to a things network application be able to send messages through any TTN gateway that it can see, even if it's owned by someone else",
then yes, it should be able to.
The gateways are effectively transparent to traffic, and just forward packets back to their router — this is one of the key advantages of TTN over a local supplier with their own gateways: "all" TTN gateways forward LoRaWAN traffic to TTN's servers.
There are some practical limitations to this though, depending on the global region, the gateways may be using a different router endpoint (you assign a regional router to the TTN application that you create), and the transmission frequencies may be different from your home region.
I've done an actual journey with a LoRaWAN device running, picked up the TTN gateways in Liverpool when I set off, and got traffic going through a gateway in Oxford as I went past on the ring road, in the same OTAA session.