For \(u: \mathbb{R}\to\) Banach space \(X\), the equation considered is \[ u'(t)= Au(t)+ F(t, u_t)\quad\text{for }t\in \mathbb{R}^+:= [0,\infty),\tag{\(*\)} \] where \(u_t:= u(\cdot+ t)|\mathbb{R}^-\), \(\mathbb{R}^-:= (-\infty,0]\), \(A\) is the infinitesimal generator of a compact semigroup \((T(t))_{t\in\mathbb{R}^+}\) of bounded linear operators on \(X\), \(F\in C(\mathbb{R}^+\times B,X)\), \(F\) is asymptotically almost-periodic (aap) in \(t\) uniformly on \(B\); the fading memory space \(B\subset X^{\mathbb{R}^-}\) is assumed to be linear complete with seminorm \(|\cdot\|_B\), closed with respect to the uniformly bounded locally uniform convergence, \(v_r\in B\) and \(r< t\) imply \(v_t\in B\), \(v_t\) is \(|\cdot|_B\)-continuous, \(c_1\|v(t)\|_X\leq |v_t|_B\leq c_2\|v_t\|_\infty\) and \(|w_t|_B\to 0\) as \(t\to\infty\) if \(w_0\in B\), \(w|\mathbb{R}^+\equiv 0\). Results: If a solution to \((*)\) is totally stable, it is aap; here, totally stable means: to \(\varepsilon\) there is \(\delta\) such that if \(|u_0- w|\leq\delta\) on \(\mathbb{R}^-\), \(h\in C(\mathbb{R}^+, X)\) with \(|h|\leq\varepsilon\) on \(\mathbb{R}^+\) and \(v\) is a solution to \((*)\) with \(F+h\) instead of \(F\) and \(v_0= w\), then \(|u-v|\leq \varepsilon\) on \(\mathbb{R}^+\). A generalization is: If the ``limiting equation'' \(v'(t)= Av(t)+ G(t, v_t)\) has a totally stable solution \(v\) with \(u(\cdot+ r_n)\to v(\cdot)\) locally uniform on \(\mathbb{R}\) and \(F(\cdot+ r_n,w)\to G(\cdot, w)\) uniform on \(\mathbb{R}^+\times B\) with some \(r_n\to \infty\), then \(u\) is aap; if the Cauchy problem for \((*)\) has at most one solution, \(u\) is also totally stable. Analogous results are obtained for stable (\(h\equiv 0\) above) and \(B\)-(totally) stable, where \(|u_0-w|_B\leq\delta\).