I present the construction of classical gauge theories based on free differential algebras and $L_\infty$ algebras. By gauging these algebraic structures, I derive extended Chern-Simons and transgression forms, as well as generalized anomaly terms. As an application, I study the kinematical algebras of relativistic, non-relativistic, and Carrollian spacetimes, and analyze how they admit consistent extensions in terms of $L_\infty$ structures.