Context. The photospheric abundances of evolved solar-type stars of different metallicities serve as probes into stellar evolution theory. Aims. Stellar photospheres of post-asymptotic giant branch (post-AGB) stars bear witness to the internal chemical enrichment processes, integrated over their entire stellar evolution. Here we study post-AGB stars in the Large Magellanic Cloud (LMC). With their known distances, these rare objects are ideal tracers of AGB nucleosynthesis and dredge-up phenomena. Methods. We used the UVES spectrograph mounted on the Very Large Telescope at the European Southern Observatory, to obtain high-resolution spectra with high signal-to-noise of a sample of four post-AGB stars. The objects display a spectral energy distribution that indicates the presence of circumstellar dust. We perform a detailed abundance analysis on the basis of these spectra. Results. All objects are C-rich, and strongly enhanced in s-process elements. We deduced abundances of heavy s-process elements for all stars in the sample, and even found an indication of the presence of Hg in the spectrum of one object. The metallicity of all stars except J053253.51-695915.1 is considerably lower than the average value that is observed for the LMC. The derived luminosities show that we witness the late evolution of low-mass stars with initial masses close to 1 M⊙. An exception is J053253.51-695915.1 and we argue that this object is likely a binary. Conclusions. We confirmed the correlation between the efficiency of the third-dredge up and the neutron exposure that is detected in Galactic post-AGB stars. The non-existence of a correlation between metallicity and neutron irradiation is also confirmed and expanded to smaller metallicities. We confirm the status of 21 μm stars as post-carbon stars. Current theoretical AGB models overestimate the observed C/O ratios and fail to reproduce the variety of s-process abundance patterns that is observed in otherwise very similar objects. Similar results have recently been found for a post-AGB star in the Small Magellanic Cloud (SMC).