Context. The dominant mechanism leading to the formation of brown dwarfs (BDs) remains uncertain. While the census of Class II analogs in the substellar domain continues to grow, the most direct keys to formation, which are obtained from younger objects (pre-BD cores and proto-BDs), are limited by the very low number statistics available. Aims. We aim to identify and characterize a set of pre- and proto-BDs as well as Class II BDs in the Lupus 1 and 3 molecular clouds to test their formation mechanism. Methods. We performed ALMA band 6 (1.3 mm) continuum observations of a selection of 64 cores previously identified from AzTEC/ASTE data (1.1 mm), along with previously known Class II BDs in the Lupus 1 and 3 molecular clouds. Surveyed archival data in the optical and infrared were used to complement these observations. We expect these ALMA observations prove efficient in detecting the youngest sources in these regions, since they probe the frequency domain at which these sources emit most of their radiation. Results. We detected 19 sources from 15 ALMA fields. Considering all the pointings in our observing setup, the ALMA detection rate was ∼23% and the derived masses of the detected sources were between ∼0.18 and 124 MJup. We classified these sources according to their spectral energy distribution as 5 Class II sources, 2 new Class I/0 candidates, and 12 new possible pre-BD or deeply embedded protostellar candidates. We detected a promising candidate for a Class 0/I proto-BD source (ALMA J154229.778−334241.86) and inferred the disk dust mass of a bona fide Class II BD. The pre-BD cores might be the byproduct of an ongoing process of large-scale collapse. The Class II BD disks follow the correlation between disk mass and the mass of the central object that is observed at the low-mass stellar regime. Conclusions. We conclude that it is highly probable that the sources in the sample are formed as a scaled-down version of low-mass star formation, although disk fragmentation may be responsible for a considerable fraction of BDs.