  We construct an $I$-family of ancient graphical mean curvature flows over a
minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the
Morse index $I$ by establishing exponentially fast convergence in terms of
$|x|^2-t$. As a corollary, we show that these ancient flows have finite total
curvature and finite mass drop. Moreover, one family of these flows is mean
convex by a pointwise estimate.
