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Primary cosmic rays produce, upon entering the Earth's atmosphere, a whole load of secondary particles. These primary particles are necessarily stable particles such as protons, electrons, and neutrinos. Wikipedia says: "When cosmic rays enter the Earth's atmosphere they collide with atoms and molecules, mainly oxygen and nitrogen. The interaction produces a cascade of lighter particles, a so-called air shower secondary radiation that rains down, including x-rays, muons, protons, alpha particles, pions, electrons, and neutrons.[64] All of the produced particles stay within about one degree of the primary particle's path."

Famously, muons produced this way can reach the Earth's surface and be detected.

Highly energetic photons above the threshold for pair production would spontaneously produce electron-positron pairs, I would guess.

My question, however is, what happens with neutrons in particular? They have a much longer half-life than muons, so they would have no problem with reaching the Earth's surface before decaying, but on the other hand they are much more massive. They are not charged, so would not be deflected by electromagnetic fields. Do they "bounce" multiple times between air molecules or other particles and never reach a target close to the primary's path?

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Secondary cosmic rays consist of many different particles, and each of them is interacts differently with the atmosphere. Dorman 2004, "Cosmic Rays in the Earth’s Atmosphere and Underground" is a great classic read about this topic and available in most libraries.

Protons, muons, and electrons are charged and can actually contribute to atmospheric ionisation. Moreover, protons, neutrons, and muons can collide with atmospheric nuclei (oxygen, nitrogen) and create more particle showers, mainly neutrons.

What affects the propagation of secondary cosmic rays?

While the muon intensity depends on atmospheric temperature, neutron intensity can depend on atmospheric humidity. Almost all particles are attenuated by air molecules, so they depend on air pressure and lose energy and intensity on their way down to the ground.

The whole process of cosmic-ray particle propagation is very complex and hardly known, so many models exist that try to mimic their behaviour. Here is a nice figure from Tatsuhiko Sato 2015 showing the altitude dependence of different particles:

cosmic ray particles with altitude

At the ground, muons can easily penetrate deep. There is a whole research field, called muon tomography that makes use of cosmic-ray muons to detect caves in mountains or pyramids.

Protons and neutrons easily reach the ground. Due to their high energy and the neutron's neutral charge, they are mostly insensitive to anything and can find their way straight to the surface. Therefore, they are used as a proxy for solar activity and galactic CR variations. At the neutron monitor data base you can find comprehensive explanations of the how's and why's.

Their collision with the soil occurs up to a few tens of decimeters deep, which creates evaporation neutrons (few MeV) that are much more sensitive to hydrogen for example. These neutrons either reflect back to the atmosphere or get thermalized in the ground, depending on soil density and water content. People are actually measuring soil moisture with this technique.

Do they "bounce" multiple times between air molecules or other particles and never reach a target close to the primary's path?

The secondary neutron energy is much too high (>100 MeV) for elastic collisions, they mainly interact inelastically. Hence, high-energy neutrons do not bounce around all the time, they actually keep quite tightly collimated to the incident path until they hit target nuclei to create evaporation neutrons (<10 MeV). From that moment on, interactions are elastical, leading to an isotropic scattering and bouncing of neutrons. But this happens already
quite close to (0 to 100 m above) the surface.

Since you asked about angular collimation of high-energy neutrons, it has been described by Nesterenok 2013 as $J(\alpha) = \exp{(-2.4\,(1-\cos{\alpha}))}$ and roughly looks like this:

Angular distribution of secondary cosmic-ray neutrons

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My question, however is, what happens with neutrons in particular?

Found this paper which estimates the effects of cosmic air showers on electronics.

Cosmic rays at sea level consist mostly of neutrons, protons, pions, muons, electrons, and photons. The particles which cause significant soft fails in electronics are those particles with the strong interaction: neutrons, protons, and pions. At sea level, about 95% of these particles are neutrons.

So, due to the small density of the atmosphere and the fact that neutrons have to hit the nuclei in order to interact with the strong interaction, the neutrons produced in the cosmic showers still end up at sea level of the earth .

BTW photons have to interact with the electric fields of the molecules in the atmosphere and the interactions will generate pairs of particle once of high enough energy, and also scatterings and degrading of energy.

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