The Feldgendarmerie did serve side by side with the Dutch civilian police in cases where Germany unfriendly persons were involved.
My dad was taken from his home by one of each when he was visiting his parents while actually being in hiding for dodging the required forced labour in Germany. He had been in hiding since 1941 but was caught in april 1944. Almost died Christmas 1944 due to starvation and exhaustion in Southern Germany. The advance of the Americans saved him in the end. Some Germans began to become more human then.
I assume the Feldgendarmerie was also keeping an eye on the Dutch police. Not all of them collaborated.
