Having recently visited the war graves of Belgium I understand that the Germans don't have a commission for the up keep of their cemeteries and therefore rely heavily on donations, however do the soldiers buried in commonwealth war cemeteries fall under the jurisdiction of the war graves commission as the seem to have a more prominent grave stone similar to that of the allied troops. It seems to me sad that The German government don't have a commission to ensure their sacrifices are always remembered as well.
I can't speak on the policy of how the responsible organizations of the Entente nations care for their war graves and those Germans that are buried there, but I can shed some light on the situation in Germany and clear up what may be a misinterpretation:
German war cemeteries abroad, from the First and Second World War as well as those for the fallen of the Franco-German War 1870/71, the First and Second German-Danish war and those fallen in colonial actions, are not cared for directly by the Federal Government, but are traditionally administered and cared for by the Volksbund Deutsche Kriegsgräberfürsorge - The People's Federation [for the] Care for German War Graves. It is not a government organization, but an association of private citizens founded in 1919. The organization tasked with erecting and caring for German war cemeteries during the war did no longer exist in the new German state, and the governmental commission formally tasked with caring for the War Cemeteries, the ZAK (Zentralnachweisamt für Kriegsverluste und Kriegsgräber) was not capable of carrying out everything that needed to be done. Thus private organizations sprung up, which, after some time, almost all united under the Volksbund, who thus de facto monopolized the care for War Cemeteries and marginalized the ZAK. While originally envisioned as a broad social movement encompassing all German citizens, nationalist and revanchist groups soon became dominant in the Volksbund, with remembrance events often dominated by their agendas and thus becoming a vessel for revanchist thoughts and a revanchist reframing of the deaths of the First World War. Left and socialist groups largely stayed away from the Volksbund and held their own remembrance events.
So this organization, already in the 1920s, was de facto the German organization tasked with care for the War Cemeteries. I don't want to go too much into detail, since this has already gone a bit too far off the road, but they worked well with the new NS-Regime. After the war, the Volksbund was reconstituted in 1946, and the British Military Government allowed a continuation of their efforts in 1948, and the Volksbund was by law of 1952 acknowledged as being responsible for German war cemeteries abroad, while those in Germany itself fell under responsibility of the Länder. The same law, in its many mutations, also states that the graves should be consistently marked by a "simple, dignified" gravestone, different tombstones are not permissible on the same cemetery.
Thus the Volksbund does act by appointment and in stead of the Federal Government to care for German war cemetaries abroad on the basis of bilateral treaties with the concerned countries. Here's a list of countries where this does apply. According to the Volksbund report, they get ~25% of their expenses covered by the Federal Government, the rest indeed comes from donations. The Bund (the German government) does pay for the costs that come from making a new cemetery, erecting appropriate gravestones and eventually necessary relocation of remains. The Länder pay for the upkeep of those cemeteries in Germany, the Volksbund does care for those abroad - and often for those in Germany, since they are employed by the Länder for that job or are acting in an advisory manner for local caretakers.
As a follow-up question, are Entente war cemeteries paid for entirely by the government?