In the zero-field-cooled exchange bias (ZEB) effect the unidirectional magnetic anisotropy is set at low temperatures even when the system is cooled in the absence of external magnetic field. La$_{1.5}$Sr$_{0.5}$CoMnO$_{6}$ stands out as presenting the largest ZEB reported so far, while for La$_{1.5}$Ca$_{0.5}$CoMnO$_{6}$ the exchange bias field ($H_{EB}$) is one order of magnitude smaller. Here we show that La$_{1.5}$Ba$_{0.5}$CoMnO$_{6}$ also exhibits a pronounced shift of its magnetic hysteresis loop, with intermediate $H_{EB}$ value in respect to Ca- and Sr-doped samples. In order to figure out the microscopic mechanisms responsible for this phenomena, these compounds were investigated by means of synchrotron X-ray powder diffraction, Raman spectroscopy, muon spin rotation and relaxation, AC and DC magnetization, X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism (XMCD). The parent compound La$_{2}$CoMnO$_{6}$ was also studied for comparison, as a reference of a non-ZEB material. Our results show that the Ba-, Ca- and Sr-doped samples present a small amount of phase segregation, and that the ZEB effect is strongly correlated to the systems structure. We also observed that mixed valence states Co$^{2+}$/Co$^{3+}$ and Mn$^{4+}$/Mn$^{3+}$ are already present at the La$_{2}$CoMnO$_{6}$ parent compound, and that Ba$^{2+}$/Ca$^{2+}$/Sr$^{2+}$ partial substitution at La$^{3+}$ site leads to a large increase of Co average valence, with a subtle augmentation of Mn formal valence. Estimates of the Co and Mn valences from the $L$-edge XAS indicate the presence of oxygen vacancies in all samples (0.05$leq delta leq$0.1). Our XMCD results show a great decrease of Co moment for the doped compounds, and indicate that the shift of the hysteresis curves for these samples is related to uncompensated antiferromagnetic coupling between Co and Mn.
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