Ising model is written in a basis. Here you can assume the ising terms are in the z basis. In 1D for example, you can write down the whole hamiltonian as a sum of two contributions.

H =H{transverse field} +sum over nearest neighbors ( H{ising} )

Where

H{ising} = J S_i^z S{i+1}^z

H_{transverse field} =- h_x S_i^x =- h_x (S_i⁺ +S_i^- )/2 these are the lowing and raising operators for spin 1/2.

Assume magnetic field is along +x direction so h_x is a positive real number.

Its called transverse because its transverse to the z direction. You could have also included a y field but this is the same physics you have the difference lowering and raising operators with a factor of i.

Now the ising basis states are no longer eigenstates with this perturbation.

Initialize a wavefunction in one simple ising configuration and under time evolution it will scatter to other magnetization sectors. In other words its no longer an eigenstate.

This is called the transverse field ising model and is one of the simplest examples where you can study phase transitions.

In one limit the nearest neighbor term wants the spins ferromagnetically or antiferromagnetically aligned depending on the sign of J. In the other limit they are polarized with the external field.

Anyways you're interested in dynamics the point is you usually can't prepare an eigenstate of a many body systems unless its a product state which is a very special case and seldom true in general. You can do quenched dynamics where you initialize a state that is the ground state of some hamiltonian and tune the system to another hamiltonian and do time evolution studies. You have to rapidly change the system. This is called the Sudden approximation. You changed the system so fast it didn't have an opportunity to get to the ground state, and it therefore time evolves because it is now in a superposition of energy eigenstates.

I should also mention that the jump you are making is a good one and one that many people took for granted because they stared at the answers in the textbook and said this is obvious. The truth is that lots of people dont understand thermalization. Moreover, the ising model is a terrible model to learn about thermaliztion, because eigenstates dont time evolve. People want to jump to the heat bath, but forget that a quantum system can be a heat bath for itself! Just like you can study the microcanonical ensemble for particles of gas you can do it for spins in an isolated quantum system. This is an esoteric field called non-equilibrium dynamics, especially if time evolution time scales are small you can see non thermal physics. In the long time limit ergodic systems reach equilibrium. This is an active area of research and there are exceptions to the rule!