The study is related to a linear time-dependent nonlocal volume-constrained diffusion problem of the form \[ \begin{alignedat}{2} u_t+ L u & = f & \text{ in } & \Omega_s, \quad t>0,\\ u(x,0) & =u_0 & \text{ on } & \Omega_s \cup \Omega_c,\\ V u & =g & \text{ on } & \Omega_c,\quad t>0,\end{alignedat} \] where \(\Omega_s \in \mathbb R^n \) is a bounded open domain and the linear operator \(V\) imposes constraints on certain volumes \(\Omega_c \in \mathbb R^n\). The nonlocal operator is defined as \[ Lu(x)= -2 \int \limits_{\Omega_s \bigcup \Omega_c} (u(y)-u(x)) \gamma(x,y)\, dy \;\;\forall x \in \Omega_s. \] The kernel function is nonnegative and symmetric. Given that nonlocal diffusion problems allow their solutions to have spacial discontinuities, discontinuous Galerkin approximations become natural choices when numerical approximations are considered. The key idea of the designed method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin method but with some nonlocal extensions. A theoretical study shows that the proposed semi-discrete scheme is \(L_2\)-stable, convergent and asymptotically compatible. Numerical tests are presented to demonstrate the effectiveness of the method.