Plus, the very means by which "transporting" is achieved in the show necessarily means that people can move while being beamed. After all, they always arrive in a slightly different pose from that they held when leaving. (Well, not in "Mark of Gideon" where Shatner doesn't have to physically move in order to get from starting set to finishing set, but apart from that...)
That a set of atoms arriving is "the same" as the set of atoms that departed is a complex issue of physics and existentialism, and not something we could ever decide on "common sense" alone. Identity becomes extremely vague at the small end of the scale, and it has e.g. been argued that there only exists one electron in the entire universe, because there could never be an identical pair (the Pauli exclusion principle in action) and thus any "other" electron could just be the one doing something different... For all we know, the atoms that left as Will Riker were the very same that arrived as two Will Rikers - there just were two of each at arrival, but still the very same.