  We explicitly compute the limiting transient distribution of the search-cost
in the move-to-front Markov chain when the number of objects tends to infinity,
for general families of deterministic or random request rates. Our techniques
are based on a "law of large numbers for random partitions," a scaling limit
that allows us to exactly compute limiting expectation of empirical functionals
of the request probabilities of objects. In particular, we show that the
limiting search-cost can be split at an explicit deterministic threshold into
one random variable in equilibrium, and a second one related to the initial
ordering of the list. Our results ensure the stability of the limiting
search-cost under general perturbations of the request probabilities. We
provide the description of the limiting transient behavior in several examples
where only the stationary regime is known, and discuss the range of validity of
our scaling limit.

Probability