I will report on recent work on the construction of anticyclotomic $p$-adic $L$-functions for Rankin-Selberg products. I will explain how by $p$-adically interpolating the branching law for the spherical pair $\left(U_n, U_n \times U_{n+1}\right)$, we can construct a $p$-adic $L$-function attached to cohomological automorphic representations of $U_n \times U_{n+1}$. Due to the recent proof of the unitary Gan-Gross-Prasad conjecture, this $p$-adic $L$-function interpolates the square root of all critical $L$-values, including anticyclotomic variation. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.