  We investigate the limiting behavior of multiple ergodic averages along
sparse sequences evaluated at prime numbers. Our sequences arise from smooth
and well-behaved functions that have polynomial growth. Central to this topic
is a comparison result between standard Ces\'{a}ro averages along positive
integers and averages weighted by the (modified) von Mangoldt function. The
main ingredients are a recent result of Matom\"{a}ki, Shao, Tao and
Ter\"{a}v\"{a}inen on the Gowers uniformity of the latter function in short
intervals, a lifting argument that allows one to pass from actions of integers
to flows, a simultaneous (variable) polynomial approximation in appropriate
short intervals, and some quantitative equidistribution results for the former
polynomials. We derive numerous applications in multiple recurrence, additive
combinatorics, and equidistribution in nilmanifolds along primes. In
particular, we deduce that any set of positive density contains arithmetic
progressions with step $\lfloor p^c \rfloor$, where $c$ is a positive
non-integer and $p$ denotes a prime, establishing a conjecture of
Frantzikinakis.
