I can’t explain the physical meaning for all $f$ but I can say that Lorentz transformations also mix time and space derivatives so in a sense this is normal. This is because the derivative $\partial^\mu$ 4-vector is well… a 4-vector so it transforms under Lorentz transformations like $x^\mu$.
What is the meaning of mixing time and space derivatives in special relativity frame transformations?
From an unnamed resource online:
Comparing to Galilean transformations, the Lorentz transformation
mixes space and time coordinates between the two frames so we cannot
arbitrarily dissociate the two types of coordinates. The basic unit in
space-time is now an event, which is specified by a location in space
and time given in relation to any system of reference. This mixture of
space and time makes it evident that we must abandon our cherished and
intuitive notion of absolute time.
With that in mind, if we take the typical 1D $x$-boost transformation (which now we are considering needs to mix space and time coordinates),
$$x' = x'(x,t)$$
Consider some function $f$ of $x'$. By the chain rule
$$\frac{\partial f(x')}{\partial t'} = \frac{\partial f(x')}{\partial x}\frac{\partial x}{\partial t'} +\frac{\partial f(x')}{\partial t}\frac{\partial t}{\partial t'}$$
and
$$\frac{\partial f(x')}{\partial x'} = \frac{\partial f(x')}{\partial x}\frac{\partial x}{\partial x'} +\frac{\partial f(x')}{\partial t}\frac{\partial t}{\partial x'}$$
so it is natural these derivatives mix when the coordinates mix (so the physical intuition follows from the quote above).
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