The short answer? Nothing, at least compared to WW2. However, power struggles in the post-war Central Powers and the fates of their heads of government is a very interesting story.

The simplest transition of power occurred in Germany, where the Kaiser went into exile in the Netherlands. However, the rest of his family retained properties in Germany and were politically active. The Crown Prince later became an adherent of the stab in the back legend and a supporter of National Socialism for a brief period. Versailles created a forced power struggle through the downsizing of the German army. Hans von Seeckt, the Chief of the General Staff for a pivotal moment in the interwar, decided to cull all officers he believed were old school or incompetent, and retain only the best talent. Still, much of the military elite kept their jobs so to speak, as the army consisted lopsidedly of officers. Post-war Germany was beset by a Communist rebellion, which was crushed with the help of the Freikorps - right wing militias made of discharged soldiers.

In Austria, the heir to the Habsburg family, Charles, who made earnest yet innocent attempts to negotiate a separate peace in the war, actually tried to rebuild his base of power in Hungary, where conditions were slightly, though not entirely, more favorable. He was blocked in this effort by Hungarian admiral Miklos Horthy, one of the top commanders of Austria-Hungary during the war. Horthy was now an admiral in a nation that had no navy, but had base of power in the remnants of the military and right wing militias. A series of crises in Hungary, related to the Treaty of Trianon and the economic disaster that followed the partition of pre-War Hungary and the war, led to governmental collapses that brought Bela Kun's Hungarian Communist Party to power. Kun launched invasions of Czechslovakia and Romania, which resulted in his government being deposed by the Romanian army.

After Kun's downfall, Horthy initiated the period of the White Terror in Hungary, a time of suppression of Communist leaders and sympathizers. In charge of the government after the Hungarian Soviet Republic, he renamed the country the Kingdom of Hungary, which implied that there had to be a "King" of Hungary. However, because the Allies had problems with Charles, the former Austro-Hungarian Emperor, being a sovereign, Horthy named himself the "Regent" of a Kingdom that, in practice, had no King. Charles died without a throne, but was later named a saint by the Roman Catholic Church - so it wasn't all bad for him. His family became politically prominent once again. The able Otto von Habsburg became an elected official in Germany, where his clan was exiled, and was one of the founders of the European Union and the institutions it was based on. Otto von Habsburg rebuilt his dynasty's reputation in Austria and Hungary, and was given a large funeral by both governments when he died. His son had less success in politics, briefly becoming an elected official in Austria before scandals undermined his career.

In Bulgaria, the bumbling Tsar Ferdinand abdicated when his nation's defeat became inevitable. His able son, Boris III, was left to rebuild the power of the Bulgarian royal house. After the war, the government was taken over by a peasant party led by Alexander Stamboliyski. In 1923, the wartime military elite led a coup against Stamboliyski that suppressed the peasant government. Some of these officers would later go on to form the Zveno, an ultranationalist military union, which reduced the Tsar to a figurehead in 1934, but the Tsar launched a counter-coup in 1935, and retook control of the government. Bulgaria was the only Central Power whose dynasty remained in power after the war.

By far the most interesting story happened in the Ottoman Empire. The Ottomans, despite being stereotyped as the sick man of Europe, performed far above expectations during the war, and were the only Central Power in late 1918 which still had its army and frontline in tact. The surrender of Bulgaria, however, meant that Constantinople was exposed, and surrender was mandatory. The nation's wartime dictator, Enver Pasha, the head of the nationalist Committee of Union and Progress (CUP), fled the nation to Germany, and remained underground. As the Russian Civil War went into full swing, Enver Pasha joined the Bolsheviks, and was welcomed with open arms. He was made a plenipotentiary of the party, and held a congress of Bolshevik officials in the Caucusus, before being dispatched to quell a Turkic rebellion in Central Asia.

This ended up being a mistake. Enver harbored lifelong ambitions of being the next great Turkic conqueror, and building a grand Turkic Empire to rival Tamerlame and Alp Arslan. Instead of suppressing the revolt, he took control of it and turned a ragtag tribal militia into a disciplined and professional fighting force, with the help of nationalist Turkish officers who joined him in Central Asia. He achieved initial success, but eventually was overwhelmed by Bolshevik reinforcements. He died at the hands of a Soviet machine gunner while charging a fortified position by himself, armed with a saber and a Koran.

However, while Enver died, the CUP did not. The Ottoman army was not yet destroyed in the war, and, angered by allied attempts to carve up Turkey, began plotting to reverse them. The Sultan, Vahdettin, long suppressed by the CUP, fired nationalist officials and appointed conservatives and religious elements. Through duplicity and political connections in Constantinople, General Mustafa Kemal, hero of Anabolu, managed to secure an appointment as Inspector-General of Ottoman armies in Anatolia. There, he controlled telegraph communications and manipulated the army to begin suppressing the Sultan's appointees, and secretly rebuilding the Ottoman army. In Anatolia, he formed an alliance with the ultranationalist General Kazim Karabekir, who was continuing World War I by himself, fighting against Armenian troops near Lake Van, in defiance of allied plans to give Eastern Turkey to Armenia.

Ethnic violence broke out throughout Turkey as ethnic Turkish militias battled with Greek occupation forces dispatched to Izmir as the precursor to a Greek land grab, and began attacking ethnic minorities. Kemal called a series of meetings of the leaders of these militias and former CUP officers and officials, first in Amasya, then in Sivas, and finally in Ankara. The Sultan's conservatives, meanwhile, signed the humiliating Treaty of Sevres, which led Kemal to declare them non-citizens of the Ottoman Empire and declare his Ankara government the legitimate authority. The CUP had been in power before the war, so it had no trouble rallying old allies in the provinces to eject the Sultan's men and take control again after the war. Kemal made agreements with France and Italy, who had deployed to Turkey, but were angered by Lloyd George's attempts to spread Greek influence in the Aegean without consulting them, essentially removing their forces from the war. Karabekir defeated the Armenians, ejecting them from the East. Kemal then focused on the Greeks, his most difficult enemy, and managed to hold them at Sakarya. The Greek government under the capable Prime Minister Venizelos collapsed due to long-standing feuds with conservatives and royalists, leading to a purge of junior officers in the Greek army by the new conservative regime. This destroyed Greek fighting capability, and enabled a Turkish counteroffensive, which swept behind Greek lines and led to a catastrophic Greek rout.

Kemal subsequently deployed his armies to the Bosporus, and sent an ultimatum to Lloyd George demanding the immediate British withdrawal from Constantinople, or else Kemal would surround and destroy the British forces there. The Lloyd George government collapsed over this crisis, and the next government agreed to a withdrawal. In the renegotiated Treaty of Lausanne, Turkey regained Izmir and Adrianople, dissolved the spheres of influence, and gained Western Armenia, which was Russian before the war. Kemal proceeded to dissolve the Sultanate - and then the Caliphate, ending the Ottoman Empire and creating the Republic of Turkey.

Though there were no war crimes tribunals after WW1, there were power struggles. In Germany, the Kaiser was made the scapegoat for the nation's defeat, and went into exile, though his Generals kept their social position. In Austria-Hungary, the Habsburg dynasty suffered a similar fate, being banished before rebuilding their political position in Germany. In Austria, a republic was formed, divorcing the country form its Habsburg past, but, in Hungary, the Austro-Hungarian admiral Horthy led the wartime Hungarian elite to suppress political rivals and create a "Kingdom" without a King. In Bulgaria, the shrewd Tsar Boris III resurrected his dynasty's fortunes by outmaneuvering his rivals in the army and civilian parties. In the Ottoman Empire, the wartime dictator Enver had a second moment of fame when he led a Turkic rebellion against the Bolshevik party in Central Asia, but was killed in the attempt. His generals went on to re-start the war in order to amend the peace terms to be more favorable to Turkey, and constituted the inner circle of the Turkish government in the interwar. The allied powers in all of this tried to prevent wartime enemies from regaining power - they were successful in the case of Charles of Austria, who could not claim his throne in Hungary because of Allied pressure, but they made no attempt to put their enemies on trial.

Sources:

Mango, Andrew. Ataturk: The Biography of the Founder of Modern Turkey

Waldman, Eric. The Spartacist Uprising Of 1919 And The Crisis Of The German Socialist Movement.

Deak, István, Endgame in Budapest.

Groueff, Stéphane. Crown of Thorns: The Reign of King Boris III of Bulgaria.

O'Kearney, Donna. Enver Pasha and the Basmachi Rebellion.

I would just add to /u/interpine's comment that what happened in the Ottoman Empire I think is even more interesting than what he alluded to.

The wartime Ottoman Empire was de facto ruled by the "Three Pashas".

After the war, the Turks themselves held courts martial dealing with war crimes by the Turkish leadership, including what is now referred to as the Armenian Genocide. The validity of these trials is somewhat disputed. Some (like Sean McMeekin) hold that they were basically show trials to appease the allies, while others (I think more rightly) recognize that, hey, the Armenian genocide did, in fact, occur, so the conclusions of the trials were probably right.

In any event, the leadership fled into exile.

Enver Pasha, as /u/interpine said, was eventually killed in battle.

But Talaat and Djemal escaped to Berlin and Central Asia, respectively.

They would eventually be assassinated by Armenians as part of Operation Nemesis, which succeeded in killing a number of the men responsible for the genocide.