The formula for a global vertical variation of a field w.r.t an infinitesimal parameter epsilon can be expressed as $$\delta_\epsilon\phi^a={\phi^a}'(x)-\phi^a(x) = \epsilon\frac {\partial\phi^a}{\partial\epsilon}$$
The formula for a local vertical variation is $$\delta_\epsilon\phi^a=G^a\epsilon+G^{a,\mu}d_\mu\epsilon$$ for some arbitrary functions $$G^a, G^{a,\mu}$$.
The first formula comes from the expansion of the transformed field, but how is the second one derived? And how it can contain a spacetime derivative since by definition of the vertical transformation we're keeping the coordinates fixed.