Growth/decay functions with time varying limits

Question

Consider a function $f(t)$ that depends on a variable, $A(t)$, such that $f(t)$ will evolve according to a growth function towards $A(t)$ if $f(t)<A(t)$ or decay exponentially towards $A(t)$ depending on if $f(t)>A(t)$. For example if $f(0)=0$ and $A(t)=A_{0}$ then $$f(t)=(1-e^{-t/\tau})A_{0}$$ and if $f(0)=A_{1}$ and $A(t)=0$ then $$f(t)=A_{1}e^{-t/\tau}.$$ Can we describe such a function if we have $A(t)$ that is a function of time?