An investigation is presented on the global boundedness for a system of nonlocal advection--diffusion equations which models a heterogeneous population over $\R^d$, $d \in \N$. For each nonlocal advection of species $i$ according to the distribution of species $j$, the corresponding convolution kernel $K_{ij}$ is assumed to have its own regularity $\nabla K_{ij} \in L^{q_{ij}}(\R^d)$, $1 < q_{ij} < \infty$.

The first a global bound is established using a Nash-type inequality to bound an energy functional. Using a graph-theoretic arguement, it is then shown that the energy bound holds when the $q_{ij}$ values have geometric mean above $d$ along every index cycle of the form $i_1 \to \cdots i_n \to i_1$ for any $1 \leq n \leq N$.

Alternatively, a sufficient smallness condition on the initial data is obtained by establishing an invairant set of the ratios between the $L^{p_i}$ norm and the mass of each species $i$ via a differential inequality.