In quantum field theories, a local transformation of a scalar field $\phi(x)$ is a transformation that involves the field and its derivatives at same point. See for instance Weinberg's QFT textbok, Vol.1 Sec. 7.7, or arxiv:hep-th/9310042, section 4.1 (it is called point transformation there).
Examples of such transformations are $\phi(x)\to \phi(x)+f(x)$ or $\phi(x)\to \int_y f(x-y)\phi(y)$, with $\hat f(p)$ analytic for small momentum $p$.
Now, under an infinitesimal dilatation $x\to (1+\epsilon)x$, the field transform as $$\phi(x)\to \phi(x)+\epsilon(\Delta_\phi+x^\mu \partial_\mu)\phi(x),$$ where $\Delta_\phi$ is the scaling dimension of the field.
Obviously, the transformation depends explicitly on $x$ in such a way that it is not local as defined above (in momentum space, it involves a derivative wrt to momentum).
My question is: how much "non-local" is the dilatation transformation? Can it be understood as local in a sense?
I ask this because integrated operators nevertheless behave nicely under this transformation, since for instance $$ \int_x \phi(x)^2\to (1+\epsilon(2\Delta_\phi-d))\int_x \phi(x)^2, $$ by integration by part.
