How did the make up of roman senete change over time?

First it's important to think about the complications involved with the word “empire.” Rome was an empire (country ruling over other countries) before the first emperor, but the word derives from imperator, the name used by Augustus. But it meant “wielder of military power,” a kind of uber-general and was specifically not supposed to connote the idea of an emperor as we think of it today (the goal was to avoid being called a king or being seen as one). Earlier, Augustus was known as dux (leader) and also, later princeps (first citizen). As far as I know, in the days of the republic, Rome called the provinces just provinciae or socii or amici, without a general term for their empire unless it was imperium romanum, but that really meant the military power of Rome (over others) without being a reference to the empire as a political entity. It didn’t become an empire because of the emperors, and the way we use these words now can cloud the already complicated political situation in Rome in the 1st century BC.

The point is this: the Roman Republic did have an empire as we conceive it, but the Senate was unwilling to make changes that would have enabled it to retain power over the empire. By leaving it to proconsuls to rule provinces, they allowed proconsuls, who were often generals of their armies whether they were actually proconsul at any given time or not, to accrue massive military power (imperium) that could be exerted over Rome itself. (This, by the way, is in part the inspiration behind moving American soldiers around so much—it takes away the long-term loyalty a soldier may have toward a particular general.)

So the Senate found itself in no position to defy Caesar, who named himself the constitutional title of dictator for increasing periods until he was dictator for life, or Octavian (later named Augustus), who eventually named himself imperator.

The Senate had plenty of warning about this. The civil wars between Sulla and Marius gave plenty of reason for it to make real changes, but they were so wedded to the mos maiorum(tradition of the ancestors) that they were not willing to address the very real dangers to the republic that their constitution, which was designed for a city-state, was facing (not that I have too many bright ideas about what they could have done).

To finally come around to the point, the Senate went from being the leading body of Rome to being a rubber stamp on whatever the imperator wished, but there was no single moment when Rome became an empire and the Senate lost power, and these transformations don't coincide.

For one thing, the second triumvirate was legally sanctioned (unlike the informal first triumvirate), so it was a temporary measure—it lasted two 5-year terms— and the time Octavian spent as dux was ambiguous as to where he actually stood or would stand over the long term (in 33 BC, the second term of the second triumvirate expired, and he was not made imperator until 27). When he named himself imperator, he solidified that relationship and took on the posts of consul and tribune (and various combinations of posts as time went on).

If we simplify, we would say that the Senate was the leading body of Rome before the first emperor and a prestigious but powerless body afterwards, though senators were influential in their own milieus.

One other thing to keep in mind is that Octavian’s rise to Caesar Imperator Augustus Was by no means peaceful and amicable. He gets a reputation in many people’s minds as dictatorial but stable and peaceful, but the proscriptions of the second triumvirate were every bit as bloody and greedy as those of Sulla. Ironically, it was Julius Caesar who was forgiving to his former enemies after he named himself dictator. Augustus did end widespread killings and confiscations after becoming imperator, but that was only after striking fear into everyone and wiping out all his enemies, including the likes of Cicero.