Classifying the compact subsets of $L^p$

Look at the nice notes by Hanche-Olsen and Holden on the Kolmogorov-Riesz compactness theorem characterizing norm compactness in $L^p(\mathbb{R}^n)$ for $1 \leq p \lt \infty$ in terms of uniform $p$-integrability and $p$-tightness, see Theorem 5 on page 3 for the precise statement of the result. This is very much in the spirit of the theorems you mention. Don't miss the historical notes in section 4, where you can find a host of references to further related results and applications. These notes and the references therein should answer your question entirely.

I'd also like to point you to the quite closely related Dunford-Pettis theorem on weak compactness in $L^1$.