The authors study a nonlocal equation of the form \(u_t = J*u -u + G*(f(u)) - f(u)\) in \((0,\infty)\times\mathbb R^d\) subject to the initial condition \(u(x,0) = u_0(x)\), \(x \in\mathbb R^d\), with \(J\) radially symmetric and \(G\) not necessary symmetric. The nonlinearity \(f\) is assumed to be nondecreasing with \(f(0) = 0\) and locally Lipschitz continuous. The nonnegative functions \(J\), \(G\) are smooth and satisfy \[ \int_{\mathbb R^d}J(x)\,dx = \int_{\mathbb R^d}G(x) = 1. \] First, the existence and uniqueness of a solution \(u \in C([0,\infty);L^1(\mathbb R^d))\cap L^{\infty}([0,\infty);L^{\infty}(\mathbb R^d))\) is proven for an initial function \(u_0(x) \in L^1(\mathbb R^d)\cap L^{\infty}(\mathbb R^d)\). Moreover, the following contractive property \[ \| u(t) - v(t)\| _{L^1(\mathbb R^d)} \leq \| u_0 - v_0\| _{L^1(\mathbb R^d)} \] holds for any \(t \geq 0\) and \(u_0\), \(v_0 \in L^1(\mathbb R^d)\cap L^{\infty}(\mathbb R^d)\). Finally, the authors establish the asymptotic behavior of solutions as \(t\to\infty\) when \(f(u) = | u| ^{q-1}u\) with \(q > 1\) and find both the decay rate of the solution obtained and the first-order term in the asymptotic regime.