Are the air particles in today's wind on earth (more or less) the same as the air 2/3 billion years ago? So the air pressure on earth remains relatively constant, right?
Is there air gained or lost through transitions of any sort? e.g. plumes going out into space, earth gaining new air particles with gravity from air in the solar system.
I was curious to know whether in general the particles that brushed up against the dinosaurs, and against Homo erectus's face, are the same as the wind today, ebbing and flowing as pressure variation directs it. Is this true? :)
I suppose air goes into trees, and into mammals, and recycles around. It seems like the recyclability rate must be very very high.
Is this logic correct? It's all very fascinating!
 A: For some atmospheric gases, this is very unlikely, but for some it is possible to a limited extent.  Each gas has its own residence time, which is the average time that

every substance released into the atmosphere is eventually removed, so that we
  have a cycle of elements.

Also known as biogeochemical cycles, from "Atmospheric Composition, Global Cycles and Lifetimes".
These can range from hours and days for air pollution species to 100's days for ozone and well over a million years for nitrogen, according to this table from Hobbs - so in this case, the same nitrogen gas we breath and brush against our face may have indeed made contact with our distant relative.
An old, yet still relevant article is "Residence time and spatial variability for gases in the
atmosphere" (Hamrud, 1983).
I hope this helps.
A: The atmosphere of the Earth is mainly composed of nitrogen (N2, 78%) and oxygen (O2, 21%) molecules, which together make up about 99% of its total volume.  The remaining 1% contains all sorts of other stuff like argon, water and carbon dioxide, but let's ignore those for now.
As you probably know, the oxygen we breathe is produced by plants from water and carbon dioxide as a byproduct of photosynthesis.  Conversely, animals (including humans) use the oxygen to burn organic compounds (like sugars, fats and proteins) back into water and carbon dioxide, obtaining energy in the process.  So do many bacteria and fungi, too, and some oxygen also gets burned in abiotic processes like wildfires and the oxidization of minerals.
The result is that oxygen cycles pretty rapidly in and out of the atmosphere.  According to the Wikipedia article I just linked to, the average time an oxygen molecule spends in the atmosphere before being burned or otherwise removed from the air is around 4,500 years.
The most recent known Homo erectus fossil dates from about 143,000 years ago, so the probability that a particular oxygen molecule hitting your face today has been around since that time is roughly $\exp(- 143000 / 4500) = \exp(-31.78) \approx 1.58 \times 10^{-14}$, i.e. basically zero.
Of course, the oxygen atoms used for respiration don't disappear anywhere: they just become part of the water and carbon dioxide molecules.  Those that end up in carbon dioxide usually get photosynthesized back into free oxygen pretty soon, unless they happen to get trapped in a carbonate sediment or something like that.  The oxygen atoms that end up in water, on the other hand, may spend quite a long time in the oceans before being recycled back into the air; if I'm not mistaken, the total amount of oxygen in the hydrosphere is about 1000 times the amount in the atmosphere, so the mean cycle time should also be about 1000 times longer.  Still, eventually, even the oxygen in the oceans gets cycled back into the atmosphere.  Thus, while the oxygen molecules you breathe might not have been around for more than a few thousand years, the atoms they consist of have been around since long before the dinosaurs.

How about nitrogen, then?  Perhaps a bit surprisingly, given how inert nitrogen generally is, it's also actively cycled by the biosphere.  Unfortunately, the actual rate at which this cycling occurs seems to be still poorly understood, which makes estimating the mean cycle times difficult.
If I'm reading these tables correctly, the annual (natural) nitrogen flux in and out of the atmosphere is estimated to be somewhere between 40 and 400 teragrams per year, while the total atmospheric nitrogen content is about 4 zettagrams.
This would put the mean lifetime of a nitrogen molecule in the atmosphere somewhere between 10 million and 100 million years, well above the time since Homo erectus first appeared (about 1.8 million years ago).  Thus, it seems that most of the air molecules around you have probably been around since the days of Homo erectus, and some of them might even have been present during the age of the dinosaurs, which ended about 66 million years ago.
A: I think the other posters may have forgotten something.
The Earth’s atmosphere is formed and replenished from volcanic emissions of water vapor, etc.
And that’s good, because it’s constantly being depleted by leaking away into Space, from friction with the Solar Wind.
I believe the calculation was that if the atmosphere wasn’t being constantly replenished, Earth would be uninhabitable in about 20 Million years.
So, on your 2/3rds Billion timescale, I’d say the atmosphere has been replaced multiple times since then.
On the Human timescale, the air you breathe blew in Caesar’s face while Rome burned.  
On the geological scale, only a trace of that original atmosphere still exists in gaseous form.
