The
$d$-process generates a graph at random by starting with an empty graph with
$n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most
$d-1$ and are not mutually joined. We show that, in the evolution of a random graph with
$n$ vertices under the
$d$-process with
$d$ fixed, with high probability, for each
$j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from
$j$ to
$j+1$ when the number of steps left is on the order of
$\ln (n)^{d-j-1}$. This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree
$j$ disappears, the number of steps left divided by
$\ln (n)^{d-j-1}$ converges in distribution to the exponential random variable of mean
$\frac{j!}{2(d-1)!}$; furthermore, these
$d-1$ distributions are independent.