Things are more complicated than your question implies.
As Sebastian Riese mentions, higher energy usually means lower cross section. There is a lot of complexity. St some intermediate energies there are resonances. It is getting far afield from your question. But let us set that aside. Let us suppose there is an interaction.
It will depend on the target nucleus. This will give you many possible results.
In the following you should take it as understood that many of these reactions can produce some extra gammas along the way. If a reaction puts a nucleus into an elevated energy state, an isomer, then it can give off a gamma to relax back to the ground state. Even with this warning I am glossing over a lot of details.
For some isotopes, it will be possible for the incoming neutron to produce fission. The result is often two fragments of the original nucleus, plus some few neutrons. The number of neutrons is random and usually in the range of 1 to 3, depending on the type of nucleus and the energy of the neutron. The fragments are very often radiocative since they will very likely have too many neutrons to be stable. There are several possible radioactive decays they can follow, depending on what isotope they are. They can release a neutron. They can release an alpha particle. They can beta decay. Some can do an electron capture. And lots of them will release gammas on the way to some other decay. Fission fragments are a "soup" of many different types of radiation at many different energies and many different half lives.
Many isotopes can capture an incoming neutron. Depending on the energy of the incoming neutron, this can happen a few ways.
If there is an energy state available for the resulting nucleus, it is possible for it to simply absorb the neutron. Much more usual is what is called an n-gamma reaction. The neutron gets absorbed and the nucleus immediately releases a gamma to allow it to go into one of its available energy states. This can result in a new nucleus with one more neutron. This new isotope can be stable or it can be radioactive. This will depend on the starting nucleus and the energy of the incoming neutron.
For an example, consider iron. Natural iron has four isotopes: Fe54, Fe56, Fe57, and Fe58. Fe56 is by far the most common. But for fun, suppose the target nucleus is Fe54, which is 5.85% of naturual iron. So it catches a neutron and becomes Fe55. Fe55 can do electron capture with a half life of 2.73 years, and become Mn55, which is stable.
There are other things the incoming neutron can do. One important reaction is called "spalling." The incoming neutron can cause neutrons from the target nucleus to get kicked off. This is called an n-2n reaction. It can happen in lead. The result is, lead can be a poor choice for shielding against neutrons since it can result in more neutrons than you started with. They will be at lower energy than you started with, but more of them. And in some situations, that is worse than no shielding at all.
As to the lead nucleus, lead has four stable isotopes: Pb204, Pb206, Pb207, and Pb208. So if the spalling moves from one to the other of these it is still stable. PB205 can electron capture (possibly after releasing some gammas) with a half life of 17.3 million years. And PB203 can electron capture to Tl203 with a half life of 51.9 hours, and Tl203 is stable.
There are some other reactions that can happen, depending on the nucleus involved. But it is already quite complicated.
So, to sum up. It depends on the energy of the incoming neutron and the type of nucleus it is hitting. It can produce a new stable isotope, or it can produce a radioactive isotope through one of several reactions. And it can release several different types of radiation.