  In this article, we focus on a doubly nonlinear nonlocal parabolic initial
boundary value problem driven by the fractional $p$-Laplacian equipped with
homogeneous Dirichlet boundary conditions on a domain in $\mathbb{R}^{d}$ and
composed with a continuous, strictly increasing function. We establish
well-posedness in $L^1$ in the sense of mild solutions, a comparison principle,
and for restricted initial data we obtain that mild solutions of the
inhomogeneous evolution problem are strong. We obtain $L^{q}$-$L^{\infty}$
regularity estimates for mild solutions, implying decay estimates and extending
the property of strong solutions for more initial data. Moreover, we prove
local and global H\"older continuity results as well as a comparison principle
that yields extinction in finite time of mild solutions to the homogeneous
evolution equation.
