This part of the play is all about the 'fine art of persuasion' - particularly,as we are all used to it, in terms of politicians getting us to vote for them. Another analogy might be the jury in a court room who are so blown away by an attorney's defense that they are ready to acquit the defendant there and then...until they hear the prosecution case. It is also about good oratory or speech-making. Both men are so persuasive that they carry the 'fickle?' people with them. Think also of Jesus and the crowd baying for his blood and asking for Barrabas.
Perhaps the people are not just 'fickle' and playwrights and audiences including us, are doing them a dis-service. Perhaps what they really need is cool detachment and objectivity so that they can weigh up, compare and contrast each character's speech before rushing off like hot-headed fools to carry out some action or vengeance on behalf of the last speaker to rouse their blood.
First, they are all for Brutus, responding so positively that they even ask him to be their king instead of Caesar. Turned again by the power of invective, they then go for the rousing words of Antony because his skill lies in 'playing with their heads and emotional drives' even to the extent of inaccuracy-perhaps deliberate. Both men manipulate the people's feelings and it is so successful that their early championing of Brutus is completely forgotten in favor of the next speaker who comes along.
Julius Caesar is a play about how language can be used to manipulate. The two speeches by Brutus and Antony immediately succeeding Caesar's assassination manipulate the crowd into condoning, then condemning the act. Shakespeare has a bit of foreshadowing at the top of the play, where the tribunes manipulate the crowd through speech from celebrating Caesar's arrival to mourning Pompey's death, and the crowd disperses. Later, Brutus' and Antony's speeches move the crowd from mourning to avenging Caesar's death, and the crowd disperses to punish the "honorable men." See more at the link: