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Aims. We aim to search and characterize inflows and outflows of molecular gasin four ultraluminous infrared galaxies (ULIRGs) at $z\sim0.2-0.3$ and onedistant QSO at $z=6.13$.  Methods. We use Herschel PACS and ALMA Band 7 observations of the hydroxylmolecule (OH) line at rest-frame wavelength 119 $\mu$m which in absorption canprovide unambiguous evidence for inflows or outflows of molecular gas innuclear regions of galaxies. Our study contributes to double the number of OHobservations of luminous systems at $z\sim0.2-0.3$, and push the search formolecular outflows based on the OH transition to $z\sim6$.  Results. We detect OH high-velocity absorption wings in three of the fourULIRGs. In two cases, IRAS F20036-1547 and IRAS F13352+6402, the blueshiftedabsorption profiles indicate the presence of powerful and fast molecular gasoutflows. Consistent with an inside-out quenching scenario, these outflows aredepleting the central reservoir of molecular gas at a similar rate than theintense star formation activity. In the case of the starburst-dominated systemIRAS 10091+4704, we detect an inverted P-Cygni profile that is unique amongULIRGs and indicates the presence of a fast ($\sim400$ km s$^{-1}$) inflow ofmolecular gas at a rate of $\sim100~M_{\odot}~{\rm yr}^{-1}$ towards thecentral region. Finally, we tentatively detect ($\sim3\sigma$) the OH doubletin absorption in the $z=6.13$ QSO ULAS J131911+095051. The OH feature isblueshifted with a median velocity that suggests the presence of a molecularoutflow, although characterized by a modest molecular mass loss rate of$\sim200~M_{\odot}~{\rm yr}^{-1}$. This value is comparable to the small massoutflow rates found in the stacking of the [CII] spectra of other $z\sim6$ QSOsand suggests that ejective feedback in this phase of the evolution of ULASJ131911+095051 has subsided.

Molecular Gas Inflows and Outflows in Ultraluminous Infrared Galaxies at $z\sim0.2$ and one QSO at $z=6.1$