lagrangian formalism - How to find the continuous transformations which leave the action invariant?

Assume one has a continuous transformation of fields, and also of coordinates - in case if we consider coordinate transformations as well. Global internal symmetries, rotations, translations, dilatations, whatever, or any other. How do we generally check whether the action is invariant under such a transformation?

I feel like my question is way more basic than Noether theorem, charges, currents, etc.

To be more precise. Let's change the fields (and probably the coordinates) as:

$$\begin{align} x &\to x + \xi(x)\\ \phi(x) &\to \phi(x) + \delta\phi(x) \end{align}$$ where $\xi(x)$ and $\delta \phi(x) = F[\phi(x),\partial\phi(x)]$ have certain functional form. I would like to know, whether the action $$\begin{equation} S = \int \operatorname{d}^D x\, \mathcal{L}[g^{\mu\nu}, \phi(x), \partial\phi(x)] \end{equation}$$ is form-invariant under such a transformation. In other words, whether it will go into something like $$\begin{equation} \tilde{S} = \int \operatorname{d}^D \tilde{x}\, \mathcal{L}[\tilde{g}^{\mu\nu}, \tilde{\phi}(\tilde{x}), \partial\tilde{\phi}(\tilde{x})] \end{equation}$$ (notice no tilde above $\mathcal{L}$! I guess, that is form-invariance...)

So far, it seems to me that the obvious option - to calculate $\delta S$ does not give anything (that's what we do when deriving Noether currents). On classical trajectories, we end up with smth like $\delta S = \partial(\ldots)$ - which if of course correct, but does not help to answer my question at all.

I feel like my question should be related to Killing vectors... In various textbooks I found how the Lie derivatives are applied to the metric in order to determine which transformations preserve it. Seems like I'm interested in a similar procedure for the action.

As usually, any references greatly appreciated.

UPDATE

Since the original question can be answered in a brute force way - "just plug your transformations into the Lagrangian and see how it goes", let me ask it in a slightly more general way:

Given the action, what is the procedure to find all continuous symmetries which leave it form-invariant?