A search for direct top-squark pair production with Higgs or Z bosons in the decay final state is performed using a data sample of proton-proton collisions at $\sqrt{s}=8$~TeV, collected by the CMS detector at the LHC that corresponds to an integrated luminosity of 19.5~fb$^{-1}$. The search is performed in a sample of events containing leptons and b jets. No evidence for a significant excess of events over the standard model background prediction is observed. The results are interpreted in the context of simplified models with pair production of a heavier top-squark mass eigenstate $\tilde{\mathrm{t}}_{2}$ decaying to a lighter top-squark mass eigenstate $\tilde{\mathrm{t}}_{1}$ via either $\tilde{\mathrm{t}}_{2}\rightarrow \mathrm{H} \tilde{\mathrm{t}}_{1}$ or $\tilde{\mathrm{t}}_{2}\rightarrow \mathrm{Z} \tilde{\mathrm{t}}_{1}$, followed in both cases by $\tilde{\mathrm{t}}_{1}\rightarrow \mathrm{t}\tilde{\chi}_{1}^{0}$. The interpretation is performed in the region of signal mass parameter space $m_{\tilde{\mathrm{t}}_{1}} - m_{\tilde{\chi}_{1}^{0}} \sim m_{\mathrm{t}}$, which so far is not probed by existing searches for direct top-squark pair production. The analysis excludes top squarks with masses $m_{\tilde{\mathrm{t}}_{2}}<575$~GeV and $m_{\tilde{\mathrm{t}}_{1}}<400$~GeV at the 95\% confidence level.

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