Does AE have effects that can make footage look like it's dirty, old? As if the footage was shot decades ago and had degraded over time? Also, are there effects that could make the footage look dirty, as if the footage was shot in a dirty (eg: desert) environment?
After Effects has a lot of effects that you can use together to create dirty and "distressed" looks. A good starting point is to use some of the
Noise & Grain effects and blend the noise layers with your original footage.
I looked at the sample that you linked to, and it seems that much of their "desert" look comes from color correction with something like the
A lot of the "old" look (the flickering) seems to come from animating (wiggling) the
There's also a fluctuating vignette that is probably just an elliptical mask with animated Scale and a high Mask Feather amount.
There are also a lot of free and commercial plug-ins and animation presets for doing this sort of thing.