  We obtain new equitightness and $C([0,T];L^p(\mathbb{R}^N))$-convergence
results for finite-difference approximations of generalized porous medium
equations of the form $$ \partial_tu-\mathfrak{L}[\varphi(u)]=g\qquad\text{in
$\mathbb{R}^N\times(0,T)$}, $$ where $\varphi:\mathbb{R}\to\mathbb{R}$ is
continuous and nondecreasing, and $\mathfrak{L}$ is a local or nonlocal
diffusion operator. Our results include slow diffusions, strongly degenerate
Stefan problems, and fast diffusions above a critical exponent. These results
improve the previous $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$-convergence
obtained in a series of papers on the topic by the authors. To have
equitightness and global $L^p(\mathbb{R}^N)$-convergence, some additional
restrictions on $\mathfrak{L}$ and $\varphi$ are needed. Most commonly used
symmetric operators $\mathfrak{L}$ are still included: the Laplacian,
fractional Laplacians, and other generators of symmetric L\'evy processes with
some fractional moment. We also discuss extensions to nonlinear
  possibly strongly degenerate convection-diffusion equations.
