It is a quite common trend in the study of composition operators between spaces of analytic functions in the unit disc of the complex plane that if boundedness corresponds to a “big-oh” condition, then compactness is determined by the “little-oh” of such condition; see, for instance, Chapter 3 in [J.H.Shapiro, “Composition operators and classical function theory” (Universitext: Tracts in Mathematics; New York: Springer–Verlag (1993; Zbl 0791.30033)]. The present article follows this trend. Given \(0<p<+\infty\) and a continuous function \(\phi:[0,1)\to (0,+\infty)\) such that for some \(0<s<t,\) \(\phi(r)=o(1-r)^s\) and \((1-r)^t=o(\phi(r))\) as \(r\to 1,\) the weighted Bergman-type space \(H(p,p,\phi)\) is the space of all analytic functions \(f\) in the unit disc such that \(\| f\| _{p,\phi}:=\int_{\mathbb D}| f(z)| ^p {\phi^p(| z| )\over{1-| z| }}\, dA(z)\) is finite, where \(dA\) is the normalized Lebesgue measure in the unit disc. The authors prove that a weighted composition operator \(uC_\varphi\) acting from \(H(p,p,\phi)\) into the Bloch space \({\mathcal B}\) is bounded if and only if \[ (1-| z| ^2)| u'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1/ p}\right) \text{ and } \]
\[ \;(1-| z| ^2)| u(z)\varphi'(z)| =O\left(\phi(| \varphi(z)| )(1-| \varphi(z)| ^2)^{1+1/ p}\right). \] The “little-oh” of such condition when \(| \varphi(z)| \to 1,\) respectively when \(| z| \to 1,\) characterizes the compactness of \(uC_\varphi: H(p,p,\phi)\to {\mathcal B},\) respectively \(uC_\varphi: H(p,p,\phi)\to {\mathcal B}_0,\) the little Bloch space. The subsequent corollaries for the Bergman spaces \(A^p=H(p,p,(1-r)^{1/p})\) are stated.