  We study the evolution of the graph distance and weighted distance between
two fixed vertices in dynamically growing random graph models. More precisely,
we consider preferential attachment models with power-law exponent
$\tau\in(2,3)$, sample two vertices $u_t,v_t$ uniformly at random when the
graph has $t$ vertices, and study the evolution of the graph distance between
these two fixed vertices as the surrounding graph grows. This yields a
discrete-time stochastic process in $t'\geq t$, called the distance evolution.
We show that there is a tight strip around the function
$4\frac{\log\log(t)-\log(\log(t'/t)\vee1)}{|\log(\tau-2)|}\vee 2$ that the
distance evolution never leaves with high probability as $t$ tends to infinity.
We extend our results to weighted distances, where every edge is equipped with
an i.i.d. copy of a non-negative random variable $L$.
