So, this is a pretty complicated question. First off, with respect to the Auschwitz Trials, it's worth noting that most of the senior leaders of the camp, including the longest-tenured commandant, Rudolf Höß, were tried in Poland in 1947; most of the accused were executed or received long prison sentences. The Frankfurt Auschwitz Trials primarily dealt with lower-level SS members who worked at Auschwitz. Similar trials took place in West Germany for personnel of other camps, including Sobibór, Treblinka, and Belzec, around the same time.

I want to note two things about the Frankfurt Auschwitz Trial, because they're illustrative for the questions you're asking: one is that there were only 22 defendants, which is obviously only a tiny fraction of the people who were responsible for the crimes at Auschwitz, and the other is that the accused were charged with specific instances of murder, rather than for acting as part of the larger criminal enterprise of the Holocaust.

The fact that only 22 of the thousands of Auschwitz personnel were charged at these trials reflects a larger truth: the vast majority of rank-and-file Holocaust perpetrators, quite frankly, got away with it. According to Günter Lewy, less than ten percent of those who were involved in Nazi crimes were ever prosecuted. Many of them were never apprehended after the war, and many others were apprehended, but were not identified as Holocaust perpetrators and were released. Some of these people simply went back to their normal lives in postwar Germany, while others settled abroad.

It's important to understand that the primary focus in the immediate aftermath of the war, especially for the Western Allies, was to arrest and try the major Nazi war criminals, not necessarily to purge the rank-and-file. Because of that prioritization of major perpetrators, many of the "smaller fish" were able to simply slip through the cracks. This includes some people who were small in terms of rank, but big in terms of their actual contributions to mass murder; for example, Erich Bauer, who operated the gas chamber at Sobibór, was captured and held as a POW, but was released and only recaptured after he was identified in a chance encounter with a Sobibór survivor in 1949. By the 1950s, the interest in prosecuting war criminals diminished to an extent, as political conditions shifted during the beginning stages of the Cold War and prosecution of Nazis became a lower priority.

I don't mean to suggest that this was simply a case of negligence or anything like that, though; in many cases, the evidence simply wasn't there to charge some of the people who were in Allied captivity. The process of gathering evidence on war criminals, particularly the low-level guys who weren't going to be tried at Nuremberg, took a very long time. Investigations continued all the way through the early 1970s, as illustrated by the records of the Bundesarchiv-Ludwigsburg, although most of the actual prosecutions and convictions took place in the 1950s and 1960s. By the late 1960s, the investigations started to become less fruitful, as, even if perpetrators were identified, many of them had died or their whereabouts were no longer discoverable, and they were never charged. In some cases, people weren't charged despite the courts having enough evidence to indict them because of procedural issues and other technicalities; this included some of the Frankfurt defendants, for whom evidence had already been uncovered in the 1950s, but who weren't brought to trial until the mid-1960s.

In addition, as I noted above, most of the people who were prosecuted for Holocaust-related crimes in postwar Germany were charged with murder, or, more often, as "accessories to murder", rather than as members of a conspiracy to commit genocide. There was a tendency to focus on individual actions rather than things people did when they were "just following orders" from above in the context of a totalitarian regime, which was a distinct trend in war crimes jurisprudence in West Germany. As a result, many of them got relatively light sentences; some were sentenced to longer terms or even life imprisonment, but many of the lower-level perpetrators only served a few years in prison, even if they had been involved in tens of thousands of deaths as extermination camp personnel or members of the mobile Einsatzgruppen death squads. This was the result of these people being charged under the penal code of West Germany, rather than under the rubric of crimes against humanity that was used at Nuremberg.

With regard to the cases of people who were only discovered decades after the war, most of these cases came about as a result of governmental reviews of historical documents and lists of known perpetrators that were then cross-checked with immigration and naturalization records. For example, in the famous case of John Demjanjuk, who was accused of being a guard at Treblinka, his name was provided to the US Immigration and Naturalization Service by a Ukrainian-American journalist (although there were suspicions that he had been fed this information by the Soviets, possibly the KGB). Many of these people came to the US or other countries after the war as displaced persons and misrepresented their wartime activities to qualify for admission and citizenship, and the truth was only uncovered many years later. In many cases, such as Demjanjuk's, these discoveries led to protracted extradition cases; some, like Demjanjuk, were extradited and faced trial, while others, such as Johann Breyer, were never extradited and died without facing trial. These trials have been controversial for several reasons, including the complicated issues with extradition, the reliability of source material (especially materials from the Soviet Union), and the legal and moral validity of trying people who are very old and in some cases already suffering from dementia. Of course, it's unlikely that there will be many more of these trials, because any surviving perpetrators would be almost 100 years old at this point.

Sources:

For the Frankfurt Auschwitz Trials and the surrounding issues, Devin O. Pendas, The Frankfurt Auschwitz Trial, 1963-1965: Genocide, History, and the Limits of the Law (Cambridge UP, 2006).

For low-level Holocaust perpetrators more generally, Günter Lewy, Perpetrators: The World of Holocaust Killers (Oxford UP, 2017).

For the Demjanjuk case and related issues, Lawrence Douglas, The Right Wrong Man: John Demjanjuk and the Last Great Nazi War Crimes Trial (Princeton UP, 2016).