The β-delayed one- and two-neutron emission probabilities (P_{1n} and P_{2n}) of 20 neutron-rich nuclei with N≥82 have been measured at the RIBF facility of the RIKEN Nishina Center. P_{1n} of ^{130,131}Ag, ^{133,134}Cd, ^{135,136}In, and ^{138,139}Sn were determined for the first time, and stringent upper limits were placed on P_{2n} for nearly all cases. β-delayed two-neutron emission (β2n) was unambiguously identified in ^{133}Cd and ^{135,136}In, and their P_{2n} were measured. Weak β2n was also detected from ^{137,138}Sn. Our results highlight the effect of the N=82 and Z=50 shell closures on β-delayed neutron emission probability and provide stringent benchmarks for newly developed macroscopic-microscopic and self-consistent global models with the inclusion of a statistical treatment of neutron and γ emission. The impact of our measurements on r-process nucleosynthesis was studied in a neutron star merger scenario. Our P_{1n} and P_{2n} have a direct impact on the odd-even staggering of the final abundance, improving the agreement between calculated and observed Solar System abundances. The odd isotope fraction of Ba in r-process-enhanced (r-II) stars is also better reproduced using our new data.