  A submonoid of the additive group $\mathbb{Q}$ is called a Puiseux monoid if
it consists of nonnegative rationals. Given a monoid $M$, the set consisting of
all nonempty finite subsets of $M$ is also a monoid under the Minkowski sum,
and it is called the (finitary) power monoid of $M$. In this paper we study
atomicity and factorization properties in power monoids of Puiseux monoids. We
specially focus on the ascent of the property of being atomic and both the
bounded and the finite factorization properties (the ascending chain on
principal ideals and the length-finite factorization properties are also
considered here). We prove that both the bounded and the finite factorization
properties ascend from any Puiseux monoid to its power monoid. On the other
hand, we construct an atomic Puiseux monoid whose power monoid is not atomic.
We also prove that the existence of maximal common divisors for nonempty finite
subsets is a sufficient condition for the property of being atomic to ascend
from a Puiseux monoid to its power monoid.
