We can read here in p. 10 (penultimate equation) that the sections on associated bundles (not necessarly vector bundle) are defined by functions $f:P\longrightarrow F$ that must satisfy the equivariance condition: $$R^{\star}_{g}f=\rho({g^{-1}})\circ f$$ where $\rho$ is the representation of $G$ on $F$ that defines the associated bundle $P\times_{\rho}F$. In turn, we define a section on $P\times_{\rho}F$ by: $$\tilde{s_{\alpha}}(m)=[(s_{\alpha}(m),(f\circ s_{\alpha})(m))]$$ where $s_{\alpha}:U_{\alpha}\longrightarrow U_{\alpha}\times G$ is a section in $P$, and $[...]$ denotes the class of equivalent elements defining the associated bundle.
I remark that:
\begin{eqnarray*} \tilde{s}_{\beta}(m) & = & \left[\left(s_{\beta}(m),(f\circ s_{\beta})(m)\right)\right]\\ & = & \left[\left(s_{\alpha}(m)g_{\alpha\beta}(m),(f\circ s_{\beta})(m)\right)\right]\;\text{(}g_{\alpha\beta}\text{are the transition functions of }P\text{)}\\ & = & \left[\left(s_{\alpha}(m),(\rho(g_{\alpha\beta}(m))\circ f\circ s_{\beta})(m)\right)\right]\\ & = & \left[\left(s_{\alpha}(m),(R_{g_{\beta\alpha}(m)}^{\star}f\circ s_{\beta})(m)\right)\right]\\ & = & \left[\left(s_{\alpha}(m),(f\circ R_{g_{\beta\alpha}(m)}\circ s_{\beta})(m)\right)\right]\\ & = & \left[\left(s_{\alpha}(m),f\left[s_{\beta}(m)g_{\beta\alpha}(m)\right]\right)\right]\\ & = & \left[\left(s_{\alpha}(m),f\left[s_{\alpha}(m)\right]\right)\right]\\ & = & \left[\left(s_{\alpha}(m),(f\circ s_{\alpha})(m)\right)\right]\\ & = & \tilde{s}_{\alpha}(m) \end{eqnarray*}