The US and Russian navies have already been transporting nuclear material all over the planet on their ships and submarines, in the form of both power-generation reactors and ballistic missiles. They have been doing so for decades with a very good safety record.

And it's unreasonable to suggest that nuclear waste couldn't be safely stored on the moon. Any future civilizations who can travel to the moon will almost certainly be acquainted with nuclear physics themselves. However, launching high-level waste into space has obvious risks of its own. It will also make it hard to recycle it later, if future technologies are able to extract more power from it.

The shorter the half-life, the more dangerous the radiation is. Radioactive elements with long half lives are generally less damaging and it's only from prolonged exposure you're in trouble.

Polonium is notoriously toxic (https://en.wikipedia.org/wiki/Poisoning_of_Alexander_Litvine...) and has a furiously short half-life. Plutonium has a half-life of about 24,000 years so the radiation is spread out over a much, much longer period of time.

You can hold a plutonium sphere and suffer no ill-effects. The same cannot be said for one of polonium, something so nasty that nanograms of the material can kill you.

So basically it's the short-lived elements you need to worry about the most. The long-lived ones are relatively harmless. The biggest concern with those is securing them from those intent on using them for harm, as refining and weaponizing nuclear waste is a concern.