Can leaking radiator hoses cause problems in the transmission? What else?

Ive got a 2007 Ford Explorer Sport Trac, had the radiator replaced last July, needed to take to a closer garage because I was afraid of way transmission was acting. They left a message saying all radiator hoses were original and basically leaking evweywhere. And there's a bunch of other problems. Something with the thermostat. They're trying to figure out if tranny is totalled. Waiting for a call back. Just wondering what problems can ensue with bad radiator hoses.

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