The gendarmerie and the carabinieri are both national police services (like the FBI). They also do things like port security.
I've seen gendarmerie, carabinieri but also local sherrifs offices; There is a world of difference between the first two and the last in terms of discipline and training. For example the first two don't make YouTube videos like these: https://youtu.be/CYunIrWDANU
I think you are slightly confused about FBI and city/county/state police forces in the US and policing in Europe. French police is a national police service like the FBI [1], Gendarmierie is a branch of military under command of the same Ministry of Interior, which also controls the National Police. It's an actual military, with military weapons, uniforms, barracks, chain of command etc. etc. There is nothing like this in the US indeed. National Guard is closest but it does not do law enforcement on regular basis.
Similarly in Italy. National Police [2] is separate from Carabinieri, who are not even under the command of Ministry of Interior, but under the Ministry of Defence, making them even more "militarized" (which makes sense, since they also serve as MP).
I've seen gendarmerie, carabinieri but also local sherrifs offices; There is a world of difference between the first two and the last in terms of discipline and training. For example the first two don't make YouTube videos like these: https://youtu.be/CYunIrWDANU