A lot of places won't allow street/track shoes on the mat or in the ring. It is definitely that way where I train at, and many of the places I've rolled at wouldn't let anyone walk across the mat with street shoes on.
The wrestling shoes are fine for doing your lifting and other normal drills. They are light, provide excellent ankle support and traction. However, they aren't made for running track because they have thinner soles and no arch support. It isn't that they can't be used to run in (a lot of wrestlers do it all the time), they just aren't anywhere near as comfortable or forgiving. If you have a history of foot or ankle problems, I wouldn't recommend running in wrestling shoes.
Given that, if you don't mind going barefoot on the mat or in the ring just go barefoot. Get a set of running shoes for when you leave to hit the track.
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