We investigate the dynamical behavior of finite rings of classical spin vectors interacting via nearest-neighbor isotropic exchange in an external magnetic field. Our approach is to utilize the solutions of a continuum version of the discrete spin equations of motion (EOM) which we derive by assuming continuous modulations of spin wave solutions of the EOM for discrete spins. This continuum EOM reduces to the Landau-Lifshitz equation in a particular limiting regime. The usefulness of the continuum EOM is demonstrated by the fact that the time-evolved numerical solutions of the discrete spin EOM closely track the corresponding time-evolved solutions of the continuum equation. It is of special interest that our continuum EOM possesses soliton solutions, and we find that these characteristics are also exhibited by the corresponding solutions of the discrete EOM. The robustness of solitons is demonstrated by considering cases where initial states are truncated versions of soliton states and by numerical simulations of the discrete EOM equations when the spins are coupled to a heat bath at finite temperatures.