Invariance of Lagrangian under active coordinate transformation This question is related to symmetry properties of the Lagrangian and conservation laws. Let us consider a one-dimensional case of a particle of mass $m$ moving along the $x$ axis such that the Lagrangian is given by $L = \frac{1}{2} m \dot{x}^2 $. Now, if there is an active coordinate transformation such that the physical location of the point mass changes from $x$ to $x + \delta x$, then this must happen in some interval of time ($t,t+\delta t$). The particle was at location $x$ at time instant $t$ and is at location $x +\delta x$ at time $t + \delta t$. Since $\dot{x}$ is in general a function of time, the Lagrangian has an implicit time dependence. Then how are we sure that the Lagrangian does not change under an active coordinate transformation?
 A: (Caveat: I'm new to this framework, and I do not pretend to be an expert.)
I think there is some confusion here about the nature of the transformation.  When you check to see if a spatial transformation leaves the Lagrangian invariant, you aren't applying the transformation over time, it gets applied instantaneously.
What you're describing (carrying out a translation over a short interval of time) sounds more like a flow.
Now, it's perfectly fine to define a flow that's carrying point $A$ to point $B$ via translation over a certain amount of time. But you would not be checking for invariance with respect to the flow, you'd be checking with respect to the tangent lift of the transformation at a fixed time $t$.  These are all spatial changes that happen instantaneously: they are time independent. You're checking for invariance at each particular time-slice $t$, but it does not play into the transformation itself.  In the case of a constant speed translation, the transformation will be identical at every time $t$ anyhow.
So to recapitulate, you can evaluate whether or not a Lagrangian is invariant

*

*With respect to a function $TM\to TM$; or

*you can also evaluate whether or not the Lagrangian is invariant with respect to (all time-$t$ maps in) a flow.

The case of a translation carried out over time should be interpreted as the second one.  Each transformation that you are checking for invariance is independent of $t$, but the family of transformations involved in the flow could be considered as indexed by time.

The matter could be kept a purely mathematical one: does the Lagrangian have any spatial dependence?

Surely, if I'm understanding you correctly. Consider the Lagrangian for gravitation of a small object with zero initial kinetic energy towards a massive object.  The only contribution to the Lagrangian then is the gravitational potential, which is weaker further out, and will yield a different value.
A: The coordinate transformations have nothing to do with physically moving the particles from one location to another. Coordinate transformations, as is evident from it's name, are transformations applied to our chosen coordinates. This has no effect on the physical reality, thus the "real" positions and the momenta of the particles remains unchanged. And, if the Lagrangian stays invariant under any such coordinate transformation, we can then exploit that symmetry to find a conserved quantity of the morion, using Noether's theorem.
In the case of active transformations, they are characterised by virtual displacements. As the wiki states, in virtual displacements, time dependence is not considered. In other words, $\delta t=0$ when talking about virtual displacements. Thus, there is no physical time elapse occuring when you perform an active transformation.
