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 $ψ$-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.