Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$. We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$, if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$, where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$. Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$, then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$.