Part VII of the United Nations Convention on the Law of the Sea details the rights and duties of nations with regard to the High Seas.
They were in international waters when their ship was seized but they had broken Russian laws in Russian waters the day before (or at the very least were willing accomplices to those breaking the laws).
Completely wrong on all counts. The oil rig is in Russia's EEZ, but is not in Russia's Territorial Waters. It's an offshore oil rig. The UN Convention on the Sea grants states jurisdiction only with regard to exploitation and preservation of resources in the EEZ; in all other regards the EEZ is treated in the same way as the High Seas (International Waters). That's the reason the activists are being held under suspicion of piracy - because it's one of the few crimes for which universal jurisdiction is recognised.
The ship was seized in the vicinity of the rig, which they never left. The entire incident occurred in the vicinity of the rig, in Russia's EEZ.
Running to international waters doesn't mean that you can get away from the country of the laws that you just broke. "International waters" isn't some sacred place where you are safe from being boarded and\or being charged for breaking the laws of the nation boarding you and many nations have recognized this in the past, usually by recognizing extended areas of responsibility outside of the legal 12 mile limit.
The Convention allows for the principal of "hot pursuit" whereby a ship which has violated a state's laws inside its territorial waters can be seized in international waters and brought back to that state
if the vessel has been continuously pursued from within territorial waters. However "hot pursuit" is not relevant in this instance as the ship at no time entered Russian territorial waters.
Under Part VII of the Convention, with the exception of piracy it is illegal for any state to board and seize any vessel or arrest the crew of any vessel on the High Seas unless that vessel flies the flag of that nation.
The Convention states that a nation which interferes with a vessel on the High Seas under the provisions of anti-piracy, without adequate cause or reason to do so, is liable to the state whose flag the vessel flies. This is why the Netherlands has filed a protest under the Convention against Russia for violating the sovereignty of a vessel flying under its flag.
Unless you want to say that the claims of Southern Ocean whale sanctuaries, that I know that you want to see protected, are completely bogus and that neither NZ or AUS have any right to claim them as such which opens a whole new kettle of fish for discussion.
The Southern Ocean Whale Sanctuary has been established by the International Whaling Commission, which is an international entity, thus the ban on whaling in the Sanctuary is international law, and not an attempt by an individual nation to extend its jurisdiction into international waters.