Non-orthogonal multiple access (NOMA) aims to increase the spectral efficiency of fifth generation (5G) networks by relaxing the orthogonal use of radio-resources. In this work, a network with multiple half-duplex (HD) buffer-aided (BA) relays is considered, where the source transmits with fixed rate towards two users. The users might demand the same rate by the source (e.g., two cellular users requiring the same service), or they could have different rate requirements (e.g., a cellular user coexisting with an Internet of Things (IoT) device). By deploying multiple BA relays, increased reliability and additional degrees of freedom are provided. Leveraging the spectral efficiency of NOMA and the increased diversity gain of BA relaying, two relay selection algorithms with broadcasting are proposed for power-domain (PD) NOMA and hybrid NOMA/OMA, namely BA-NOMA and BA-NOMA/OMA, respectively. BA-NOMA can improve the performance in terms of outage probability when the power allocation factor α is selected such that robustness against channel uncertainties due to, e.g., outdated channel state information (CSI), is provided. Moreover, BA-NOMA/OMA further improves the sum-rate by switching to OMA when the relays cannot serve the users through NOMA. For both cases, a theoretical analysis for the outage probability is conducted and the asymptotic performance is studied. Finally, numerical results and comparisons with other state-of-the-art algorithms are provided for the outage probability, average throughput and average delay.