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You could risk a VERY thin coat of spray varnish and hope for the best, but instead, why not just play them without varnish. Don't worry about it! Have fun. IF anything happens (and it probably won't) you can touch up later and varnish when you have lots of time.
You can handle models after a few hours. However they will still be mildly sticky. Don't let them touch each other or anything which tears easily. Tissue paper will bond to them and then tear. The soft plastic that comes in blister packs would be a good way of transport.