Let \(dA\) be the normalized Lebesgue measure on the unit disk \(\mathbb{D}\) in the complex plane. For a positive Borel measure \(\mu\) on the boundary \(\partial \mathbb{D}\) of \(\mathbb{D}\) denote \(\mathcal{P}_{\mu}(z)=\int_{\partial \mathbb{D}}\frac{1 - \vert z\vert^{2}}{\vert \zeta - z\vert^{2}}\,d\mu(\zeta)\), \(z\in\mathbb{D}\). Let \(H({\mathbb{D}})\) be the space of all analytic functions on \(\mathbb{D}\). The Dirichlet type space \({\mathcal{D}}_{\mu}\) consists of all functions \(f\in H({\mathbb{D}})\) with the finite norm \(\Vert f \Vert_{{\mathcal{D}}_{\mu}}=\vert f(0)\vert +\left(\int_{\mathcal{D}}\vert f'(z)\vert^{2}{\mathcal{P}}_{\mu}(z)\, dA(z)\right)^{1/2}\). The \(\alpha\)-Bloch space \({\mathcal{B}}^{\alpha}\) (\(0 < \alpha < \infty\)) consists of all functions \(f\in H({\mathbb{D}})\) with the finite norm \(\Vert f \Vert_{{\mathcal{B}}^{\alpha}}=\vert f(0)\vert +\sup_{z\in{\mathbb{D}}}\left((1 - \vert z\vert^{2})^{\alpha}\vert f'(z)\vert \right)\). The little \(\alpha\)-Bloch type space \({\mathcal{B}}_{0}^{\alpha}\) is the subspace of \({\mathcal{B}}^{\alpha}\) consisting of all functions \(f\) such that \(\lim_{\vert z\vert \to 1-0}\left((1 - \vert z\vert^{2})^{\alpha}\vert f'(z)\vert \right) = 0\). For an analytic map \(\varphi:{\mathbb{D}}\to {\mathbb{D}}\) the composition operator \(C_{\varphi}\) is defined on \(H({\mathbb{D}})\) by the formula \((C_{\varphi}f)(z) = f(\varphi(z))\), \(z\in \mathbb{D}\). Let \({\mathcal{C}}_{{\mathcal{B}}^{\alpha}}({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha})\) denote the closure of the space \({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha}\) in the space \({\mathcal{B}}^{\alpha}\). There is obtained a characterization of the space \({\mathcal{C}}_{{\mathcal{B}}^{\alpha}}({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha})\). There are obtained criteria of boundedness and compactness of operators \(C_{\varphi}:{\mathcal{B}}^{\beta} ({\mathcal{B}}_{0}^{\beta})\to {\mathcal{C}}_{{\mathcal{B}}^{\alpha}}({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha})\) and \(C_{\varphi}: {\mathcal{C}}_{{\mathcal{B}}^{\alpha}}({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha})\to {\mathcal{C}}_{{\mathcal{B}}^{\alpha}}({\mathcal{D}}_{\mu}\cap{\mathcal{B}}^{\alpha})\).