Problem: If the past-side of the portal opens into the same timeline as the present-side, then the entire timeline could be overwritten as soon as someone is sent through.
If you're worried about an "overwriting" model, as opposed to changing the past just creating a new parallel timeline which coexists with your own, then the answer is: Nothing. According to chaos theory, there's absolutely nothing you could send through that wouldn't be overwhelmingly likely to radically change the subsequent history of that world millions or billions into the future.
The reason has to do with the so-called "butterfly effect", also known as "sensitive dependence on initial conditions." The butterfly effect gets its name from the idea of re-running history twice starting from initial conditions that are nearly identical but differ in some very small way, like whether a butterfly flaps or doesn't flap at that moment, or even just a shift in the trajectory of a few air molecules. Since the weather is a chaotic system, this very small difference between the initial conditions of the two histories will lead to greater and greater divergence in the weather conditions of the two histories, until about two weeks or so later the weather conditions will be basically completely unrelated (aside from what commonalities you can predict just knowing about the average weather at that time of year, i.e. the climate). For example, in one history there may be a major hurricane hitting the east coast of America two weeks after the initial state, in another history there may be no hurricane at all two weeks after the initial state.
And it's been shown that the systems that undergo repeated collisions, including air molecules, are themselves sensitive to extremely small differences in gravity. According to this page, a mathematician calculated that a difference in initial conditions as small as the presence or absence of a single electron 100,000 light years away would lead to high levels of unpredictability in the trajectories of air molecules, or even billiard balls colliding on a frictionless surface, after 100 collisions or less:
Now, those of you who never had time for billiards because of homework will immediately say: "This is absurd." After all, the edge of our galaxy is said to be about 100,000 light-years away. Not only that, an electron is unimaginably small - about one thousand billion, billion, billionth of a kilogram. In fact, celebrated mathematician Michael Berry of England calculates that the angle through which the cue-ball would typically be deflected by the distant electron's gravity would be so small that it would be represented very roughly as a fraction of a degree by a decimal point followed by 100 zeros, then a one. Such a minuscule deflection is impossible to measure. Yet, if snooker balls kept rolling for more than a minute after being struck, its effect would become profound.
And this is the nub of chaos theory. Little things mean a lot. Incredibly minute uncertainties in the initial state of a system lead to total uncertainty in our best conceivable predictions of the futures of those systems. For instance, if you knew more or less a point in space that marked the edge of the solar system at its birth, it would be impossible, using that slightly inexact information to plot where the edge is now. It would therefore never be possible to predict the weather accurately beyond a few weeks, or the behaviour of the planets more than a few hundred million years from now (that may seem like a long time to humans, but it is a trivial amount of time astronomically and geologically). The theorists tell us it is because minute effects may grow exponentially in such non-linear systems. While this is difficult to explain in a simple way for the weather or the solar system, the snooker illustration is far simpler to grasp. It is all because of the way snooker balls bounce off each other. Because of the balls' curved ivory-like surfaces, the incredibly tiny effect of the electron at the edge of the galaxy is multiplied by 10 every time the balls collide, according to Dr. Berry. So after 101 collisions, the decimal point I referred to earlier will have marched 101 positions to the right and the effect of the distant electron will have grown to 1 degree. One more collision means a 10- degree change in direction and two more gives 1,000 degrees, an enormous effect in a short time. Snooker balls, of course, don't run for a couple of minutes and the effect is therefore hypothetical.
However, there is an essentially frictionless form of snooker taking place all around us in the air we breathe. Imagine dancing oxygen molecules to be minute snooker balls, unceasingly bouncing off each other, forever moving in new directions. Such imaginary molecular snooker balls would have incredibly tightly curved surfaces, leading to an error amplification factor of roughly 600, instead of 10 for real snooker, per molecular collision, according to Dr. Berry. He therefore calculated that after about 50 collisions, which occur in less than a blink of an eye for molecules, not taking into account the gravitational pull of one electron at the edge of the galaxy would lead to enormous error in estimating what directions an oxygen molecule went. So even assuming a purely classical Newtonian world, as Dr. Berry did to make a point, we see that there is an extraordinary and unavoidable unpredictability all around us. This is the meaning of chaos. When the limits to predictability of the smallest particles of matter are added to this, it is remarkable that on a large scale there is so much order.
So, sending a single particle through the portal from the future to the past is going to totally randomize the day-to-day weather patterns starting a week or two afterwards, assuming the past can be changed at all (i.e. assuming the universe doesn't obey the Novikov self-consistency principle). And of course, this is going to have effects on which individual animals live and die, along with where they happen to be standing at different moments which affects which of their DNA molecules happen to get struck by photons and other particles that can cause mutations. So, in the long term, evolution should be totally changed.
Very small differences in the initial conditions of small bodies like asteroids and comets also lead to unpredictability on scales of a few thousand years--for example, Fig. 1 on the second page of this paper graphs 11 simulations of the changing orbital size of a comet, and finds that with only small deviations of the initial conditions--one part in a million--they get "gross divergences of trajectories" in less than 10,000 simulated years. So any tiny gravitational perturbation 2 billion years ago is going to very likely change which asteroids and comets end up hitting the Earth over the next 2 billion years, including especially important ones like the one that's thought to have hit Earth 65 million years ago and killed off the dinosaurs.
