  Optically-compact star-forming galaxies (SFGs) have been proposed as
immediate progenitors of quiescent galaxies, although their origin and nature
are debated. Were they formed in slow secular processes or in rapid
merger-driven starbursts? Addressing this question would provide fundamental
insight into how quenching occurs. We explore the location of the general
population of galaxies with respect to fundamental star-forming and structural
relations, identify compact SFGs based on their stellar core densities, and
study three diagnostics of the burstiness of star formation: 1) Star formation
efficiency, 2) interstellar medium (ISM), and 3) radio emission. The overall
distribution of galaxies in the fundamental relations points towards a smooth
transition towards quiescence while galaxies grow their stellar cores, although
some galaxies suddenly increase their specific star-formation rate when they
become compact. From their star formation efficiencies compact and extended
SFGs appear similar. In relation to the ISM diagnostic, by studying the CO
excitation, the density of the neutral gas, and the strength of the ultraviolet
radiation field, compact SFGs resemble galaxies located in the upper envelope
of the SFGs main sequence, although yet based on a small sample size. Regarding
the radio emission diagnostic we find that galaxies become increasingly compact
as the starburst ages, implying that at least some compact SFGs are old
starbursts. We suggest that compact SFGs could be starburts winding down and
eventually crossing the main sequence towards quiescence.
