© 2025 by the authors.In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order (Formula presented.) of the temporal derivative to that of a Caputo fractional type and found that, for (Formula presented.), this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order (Formula presented.) on the spatial derivative. Here, depending on whether this order was below or above the harmonic value (Formula presented.), we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.