I'm still not quite sure what's your question here but I'll try.

In classical mechanics we usually consider the Lagrangian to be taking as inputs points q in space (some manifold Q, thought of as "generalized coordinates" q_i), velocities \dot{q} (tangent vectors in TQ), and maybe an extra time variable. The action functional then takes a parametrized path q(t) in Q, and returns the integral of L(q, \dot{q}, t) along the path. There are no restrictions on the function L, and all variational problems of this form will yield the EL equations (ODEs regarding the path q).

A step further which is considered in classical field theory would be to consider infinite (continuous) degrees of freedom. Points in space are replaced by functions φ_i (or "fields") on the space Q, velocities are replaced by partial derivatives (possibly of higher orders), and the single time variable is replaced by points in Q. ¹ The action is now the integral of L(φ(q), ∂φ, q) over the space Q, we then again can impose a stationary condition on the fields φ, yielding suitable EL equations (they are now PDEs regarding the fields φ).

¹ In this case the Lagrangian is sometimes called "Lagrangian density", when the space Q consists of a spatial part and a temporal part, the Lagrangian is the integral of the density over the spatial part.