Background: β-delayed neutron emission and fission are essential in r-process nucleosynthesis. Although the number of experimental studies covering r-process nuclei has recently increased, the uncertainties of β-delayed neutron emission and fission are still large for r-process simulations. Purpose: Our aim is to introduce a theoretical framework for the description of β- delayed neutron-emission and fission rates based on relativistic nuclear energy density-functional and statistical models and investigate their properties throughout the nuclide map. Methods: To obtain β strength functions, the relativistic proton-neutron quasiparticle random-phase approximation is employed. Particle evaporations and fission from highly excited nuclear states are estimated by the Hauser-Feshbach statistical model. β-delayed neutron branching ratios P_n are calculated and compared with experimental data, and the β-delayed fission branching ratio P_f are also assessed by using different fission barrier data. Results: Calculated P n are in a good agreement with the experimental data and the root mean square deviation is comparable to results of preceding works. It is found that energy withdrawal by β-delayed neutron-emission sensitivity varies P_n , especially for nuclei near the neutron drip line. P_f depend sensitively on fission barrier data. It is found that not only the barrier height but also the number of barrier humps is important to evaluate P_f . Conclusions: The framework introduced in this work provides an improved theoretical description of the β-delayed neutron emission and fission. Since P_f as well as P_n depend strongly on fission barrier information, four kinds of fission barrier data are used in this work to allow further sensitivity studies of the r-process nucleosynthesis on the nuclear fission. More studies on fission barrier are highly requested to assess the role of β- delayed fission in the r-process study. A complete set of calculated data for β-delayed neutron emission and fission are summarized as a table in supplemental material for its use in r-process studies as well as to complement a part of nuclear data in which no experimental data are available.