A systematic study of the total $beta$-decay half-lives and $beta$-delayed neutron emission probabilities is performed. The $beta$-strength function is treated within the self-consistent density-functional + continuum-QRPA framework including the Gamow-Teller and first-forbidden transitions. The experimental total $beta$-decay half-lives for the Ni isotopes with $Aleq$76 are described satisfactorily. The half-lives predicted from $A$=70 up to $A$=86 reveal fairly regular $A$-behaviour which results from simultaneous account for the Gamow-Teller and first-forbidden transitions. For $Zapprox$ 28 nuclei, a suppression of the delayed neutron emission probability is found when the $N$=50 neutron closed shell is crossed. The effect originates from the high-energy first-forbidden transitions to the states outside the $Q_{beta} - S_n$-window in the daughter nuclei. PACS numbers: 23.40.Bw,21.60.Jz,25.30.Pt,26.30.+k
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