Topological setting

Say we have a fiber bundle: $p: X \rightarrow B$. Let $s: B \rightarrow X$ be a section. Then $\pi_1(B,b)$ acts on $\pi_1(F_b,s(b))$ (where $F_b:=p^{-1}(b)$) by: Let $\gamma \in \pi_1(B,b)$ and $\delta \in \pi_1(F_b,s(b))$. Deform $\delta$ above $\gamma$ in such a way that above $\gamma(t)$, $\delta(t)$ will be a loop with base point $s(\gamma(t))$. At $t=1$ we will arrive at some new loop in $\pi_1(F_b,s(b))$. This defines the action of $\gamma$ on $\delta$.

Question

Is there a way to define this algebraically? Meaning: Let $p:X \rightarrow B$ be a map of integral schemes (varieties if you prefer?). Assume it is flat and with isomorphic fibers. Let $s:B \rightarrow X$ be a section. Can we then define, similarly, a monodromy action of $\pi_1(B,b)$ ($b$ a geo. point) on $\pi_1(X,s(b))$?

And maybe my setting is off. What is the right setting for this (algebraically), and how would you define this action?