Let $R=\oplus_{n\in\mathbb{N}_0}R_n$ be a homogeneous Noetherian ring with local base ring $(R_0,{\fm}_0)$ and that $M$ be a finitely generated graded $R$-module. Let $H_{R_+}^i(M)$ be the $i$-th local cohomology module of $M$ with respect to $R_+:=\oplus_{n\in\mathbb{N}}R_n$. Let $t$ be the least integer $i$ for which $H_{R_+}^i(M)$ is not Minimax and let $s$ be the most integer $i$ such that $H_{R_+}^i(M)$ is not Minimax. We prove that $H_{{\fm}_0R}^j(H_{R_+}^t(M))$ for $j=0,1$, $H_{{\fm}_0R}^j(H_{R_+}^s(M))$ for $j=d,d-1$ where $d=\Dim R_0$, and $H_{R_+}^s(M)/{{\fm}_0H_{R_+}^s(M)}$ are Artinian. Additionally, we show that $\Ass_{R_{0}}(H_{R_+}^t(M)_n)$ is asymptotically stable for $n\longrightarrow{-\infty}$.
