I will split my answer into two parts, simply because the political system Rome gradually shifted from what modern historians have termed the "principate", where the emperor was the first among equals, and simply held a very large number of legal titles on their person, and the "dominate", where the emperors gradually gained a divine and more recognisably monarchical character, begun with Diocletian's association with Jupiter and later with the Constaninians' association with Christianity.

The first answer is simply the sheer instability of the Roman political system. The donative payments, made by most emperors on the date of their ascension were quite literally large cash bribes meant to buy over the support of the legions. As happened almost every time the emperor was weak or weakened, the legions could and often did elect their own emperors. Moreover, it's also because Rome was a fundamentally legalistic society. The "emperor" during the principate had all that power, because, from a legal point of view he concurrently held almost all of the high offices (Pontifex Maximus, Tribune, etc.). The Romans recognised that their political system had a certain "impersonal" quality to it. Yes, personal wealth, prestige, and power were required to become and remain emperor, but it was a position that outlived its holder. Nobody wanted war, so it was almost always easiest to pass the purple onto male relatives of the former emperor. However, the moment the position was taken by someone weak, they were replaced. Dynasties in the principate could legitimate themselves to a certain extent, usually by co-opting the prestige of their antecedent, but again, regardless of dynastic connections, once the sharks circled, there was nothing that could be done. This is why so few emperors died naturally. Anthony Kaldellis, in writing about the Byzantine political system, a progression of the Roman one, described it as a system that constantly tested for weakness, a description equally applicable to the Principate.

Now the dynasties of the dominate were actually fairly well established and actual family units. The Constantinian dynasty all flowed directly from Constantius Chlorus, the "cousins" Julian and Gallus, were simply other grandchildren of Chlorus. Diocletian had begun this process, but Constantine took the prestige and cache of the imperial position to new heights, and instilled the need to maintain this "position" into his children, which included a continuation of the divine association of living emperors. A fairly short interregnum by Jovian saw the emergence of the Valentinianic dynasty, which actually intertwined with the Theodosian dynasty forming a proper Valentinianic-Theodosian dynasty. Valentinian and Valens, both brothers ruled each half of the empire, and were succeeded by Valentinian's sons, with his daughter marrying Theodosius whose sons and daughters produced the next generation of emperors, Valentinian III, Arcadius, and Honorius. Only Honorius produced an heir, Theodosius II where the dynasty for all intents and purposes fizzled out. These dynasties were far more institutionalised and legitimate than in the principate. This legitimacy can be observed in their use as figureheads by the many powerful generalissimos, who often worked around existing dynasties, and tried to marry into them instead of simply replacing them.

Long answer, sorry. But in summary, in the principate, it's because the political system was so unstable and volatile, that the moment a stroke of bad luck hit a dynasty and ensured a weak emperor took the throne, they would be, in all likelihood replaced. During the dominate however, they did genuinely succeed in establishing dynasties with significant legitimacy that enabled the ascension of weak, and often but not always, child emperors. Hope this answers your question!