Was there ever a series of leaders who passed their positions to the most capable candidate?

It seems strange to start such a list not with Alexander the Great. According to Krateros his last words were answering the question of who shall go command then he answered "to the strongest". Not arguing over the veracity of this nice excuse for his generals to find out what that means via fighting…

The big list for this is presented by things like the Holy Roman Empire or the Polish Kingdom. Generally: things called "elective monarchies." These constructs look like those who used them were apparently aware that reliance on genetic familiarity is not reliable in any way. An early example of the nature nurture debate influencing politics?

That makes the list on Wikipedia (which I would start with the Papacy):

Macedon, Rome / Byzantium, Holy Roman Empire, Kingdom of Jerusalem, Scandinavia, Poland-Lithuania, Bohemia, Venice, Dutch Republic, Sovereign Military Order of Malta

Most republics, democracies, Soviets would qualify as well. Stalin's son could not be king.

Of course the list is long, and there are exceptions. North Korean Kims come as easily to mind as North-American Bushes…

Examples without elections would then be phalangist Franco giving way to king Juan Carlos of Spain, Hitler designating Dönitz and Schwerin von Krosigk, Kemal to Inonu, Dollfuß to Schuschnigg, and Pilsudki to Smigly-Rydz, and for a change the good tyrant Agathocles of Syracuse, who called a popular assembly before his death and gave back democarcy to the people (well, at least in wording, it was compllicated and we lack proper sources to evaluate intentions and real effect (Berve, 1957 (PDF))

No dictatorship, so one believed, would continue beyond the death of the "tyrant," unless he managed to turn his rule into hereditary monarchy. Against this, however, it has been suggested recently that modern, totalitarian dictatorship, with its lieutenants and the help of the organized machinery of its party, has found the means to overcome the old dilemma. Where the totalitarian movement is effectively organized its existence provides for the mass interest in, as well as the leadership reservoir for, successorship into the office of the defunct leader. Continuation of dictatorial regimes in Russia after Lenin's death, in Turkey after the death of Kemal, in Austria after the assassination of Dollfuss, in Poland after the death of Pilsudski, are referred to as evidence of an improved technique and a changed situation.
_{–– John H. Herz: "The Problem of Successorship in Dictatorial Régimes; A Study in Comparative Law and Institutions", The Journal of Politics, Vol. 14, No. 1, 1952, pp. 19-40.}

But apart from these examples, there is a fundamental problem here. The question seeks to ascertain a pattern of rulership that is violation of a certain framework: legitimacy of rule.

A hereditary monarch gives his seat to his son. An elected leader is replaced by another elected leader. Since both types are excluded by the questions definition we are left with charismatic leadership.

As Max Weber already explained in The Three Types of Legitimate Rule, such a scheme is the most difficult to achieve and as we have seen in cases like Cromwell and Napoleon, transformation of this type of rule is the much preferred option. Real examples for Charismatic authority – Designation by original leader are extraordinarily hard to find,

Because the authority is concentrated in one leader, the death of the charismatic leader would constitute the destruction of the government unless prior arrangements were made. A society that faces the end of their charismatic leader can choose to move to another format of leadership or to have a transference of charismatic authority to another leader by means of succession.

And a designation of a successor already undermines the original leader's uniqueness to a large degree – if it doesn't present a real sword of Damocles again for the 'original'. This designation may be kept under wraps to avoid the problems of that (North Korea as example again), but the successor is then forced into a much weaker position with less inherited charisma.

If people start to believe in hereditary 'gentilic' qualities, a hereditary charisma becomes automatically some form of traditional legitimate rule.

After such types of Moses, Jesus, Mohammed, Alexander, Caesar, Napoleon, Hitler, any potential successors will inherit one thing for sure: a difficult time building their own charisma and basing their own rule on that alone, while still being able to transfer this model to a chosen successor of their own while maintaining some stability. Rivals and ordinary people are too quickly dissatisfied.

Thus, given the premise of "longer-lasting civilisation based on designated successors" that wants to exclude any form of election and models like adoptive Roman emperors will need to be rather short-lived? As a principle, it just doesn't work very well.