I have noticed the Dutch have sold another two of their fairly new frigates to Belgium so that they can build several OPVs, thereby keeping their shipyards busy. The Dutch have been sending a frigate to patrol around their dependencies in the Caribbean, similar to the British. In the future the Dutch intend to send an OPV to the Caribbean, a frigate is a bit of overkill.
The British have built a slightly larger River class OPV, HMS Clyde, to patrol around the Falklands. While OPVs are not frigates, they are adequate for their intended missions, freeing up frigates for more military missions. Several OPVs can be built for the price of one frigate, and with smaller crews OPVs are cheaper to operate.
Whether having a dedicated Coast Guard is up to each nation. Some do and some don't. Usually nations that have a developed coast line do have a Coast Guard, but you'll notice that many nations that don't have a developed coast line don't. I wouldn't consider Australia's or New Zealand's coasts developed. In fact, the Australians have stationed most of their patrol boats on their most undeveloped coast, their northern coast.