We consider the time-dependent fractional $p$-Laplacian equation with parameter $p\gt 1$ and fractional exponent $0\lt s\lt 1$. It is the gradient flow corresponding to the Gagliardo fractional energy. Our main result is the asymptotic behavior of solutions posed in the whole Euclidean space, which is given by a kind of Barenblatt solution whose existence relies on delicate analysis. We will concentrate on the sublinear or “fast” regime, $1\lt p \lt 2$, since it offers a richer theory. Fine bounds in the form of global Harnack inequalities are obtained as well as solutions having strong point singularities (Very Singular Solutions) that exist for a very special parameter interval. They are related to fractional elliptic problems of nonlinear eigenvalue form. Extinction phenomena are discussed.