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The aim of this paper is to provide new stability results for sequences ofmetric measure spaces $(X_i,d_i,m_i)$ convergent in the measuredGromov-Hausdorff sense. By adopting the so-called extrinsic approach ofembedding all metric spaces into a common one $(X,d)$, we extend the results ofGigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies andcompactness theorems in the whole range of Sobolev spaces $H^{1,p}$, includingthe space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. Inaddition, building on the results of Ambrosio-Stra-Trevisan, we provide localconvergence results for gradient derivations. We use these tools to improve thespectral stability results, previously known for $p>1$ and for Ricci limitspaces, getting continuity of Cheeger's constant. In the dimensional case$N<\infty$, we improve some rigidity and almost rigidity results by Kettererand Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, inthe class of $RCD(K,\infty)$ spaces we provide stability results for Hessiansand $W^{2,2}$ functions and we treat the stability of the Bakry-\'Emerycondition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarilyconstant.

New stability results for sequences of metric measure spaces with uniform Ricci bounds from below