Although the trip made from 2014 to Egypt has not been fully resolved, following the stop in Italy the trio travels back to Troy and becomes involved in the Trojan War. They probably had no significant impact on the events of the war, but it is a very dangerous situation temporally–the fact that Sherman is involved on the battlefield might mean changes in who lives and who dies even if he kills no one, which raises our genetic problem, the problem of who will be born in the future. One might think that events in 1184 B.C. are so far removed as to have little consequence in the present, but as it was with Tut the reverse is the case: because family trees become ever more interlaced over time, the impact of a single death or survival could easily change the entire population of the world. Yet since that did not happen, we can assume that there was no such change.
Our problem arises because Mr. Peabody gets left in the past, presumed dead. Sherman and Penny do not know what to do about this–but it occurs to Sherman that the solution might be to travel to the future to a moment before Mr. Peabody leaves for the past and ask him what to do. This is disastrous, although perhaps not for the reasons we see in the film. Sherman has been warned that he should not travel to a time when he existed, but in this case he decides it is worth the risk.
Meanwhile, Mr. Peabody builds another WABAC (out of improbable components) and catches up a few minutes later.
Because Sherman and Penny arrive in the future first, we feel as if that is as it ought to be: Mr. Peabody needs a bit of time to build his machine, and so Sherman and Penny leave from the past and arrive in the future, and then after they (in some sense) have arrived in the future Mr. Peabody leaves from the past and catches up with them in the future. This, though, is not thinking clearly temporally. From the time traveler’s perspective a move forward in time is a bit like fast-forwarding history, and that’s where we hit our confusion.
Let us suppose that it takes Mr. Peabody five years to build a new time machine. That means Sherman departs from 1184 B.C. and Peabody departs from 1179 B.C. Yet because Sherman is moving forward in history, he could in theory have as easily jumped from 1184 to 1179 to the moment when Peabody’s machine is ready to go–and in fact there is a sense in which he did that, moving to 1179 and beyond as he heads for 2014. Those five years which zoom past for Sherman take five years for Peabody–but all the same, when Sherman reaches 1179, Peabody is leaving for the future. They thus in some sense leave 1179 at the same time. The only logical reason for Mr. Peabody to arrive after Sherman is that he set the controls of his makeshift time machine to do so.
The question is why he would have done that, and the answer is not exactly obvious but comes with a bit of thought.