  We define a new class of enumerative invariants called $k$-leaky double
Hurwitz descendants, generalizing both descendant integrals of double
ramification cycles and the $k$-leaky double Hurwitz numbers introduced in
previous work of Cavalieri, Markwig and Ranganathan. These numbers are defined
as intersection numbers of the logarithmic DR cycle against $\psi$-classes and
logarithmic classes coming from piecewise polynomials encoding fixed branch
point conditions. We give a tropical graph sum formula for these new
invariants, allowing us to show their piecewise polynomiality and a
wall-crossing formula in genus zero. We also prove that in genus zero the
invariants are always non-negative and give a complete classification of the
cases where they vanish.
