Certain biomolecular functions require the accurate choreographing of binding events among several partners. Some examples are enzymatic reactions involving multiple sequential steps, or the DNA search for the target site performed by transcription factors. Such molecular oscillatory machines need a core protein oscillator that alternates between conformational substates to coordinate sequential binding events to partners. A productive oscillatory cycle of such a system requires the exchange between the two binding-competent conformations to occur quickly while the oscillator resides in both states for sufficient time as to facilitate binding to each target and thus ensure the right sequence of steps. The efficiency of these machines is determined by the number of productive cycles generated per unit of time. However, it is currently unclear whether achieving this type of binding choreography is feasible without coupling the process to an additional energy source. Here we investigate theoretically the technical requirements for designing the core oscillator of these machines. We looked at the oscillatory binding patterns emerging from the thermal fluctuations of a protein domain that is flexible or marginally stable, and which (un)folds in either two-state or downhill fashion. We find that the downhill scenario speeds the transition between conformational alternants, but makes the residence times in the binding-competent substates fleeting. The two-state scenario provides ample time for binding in either substate, but makes the transitions too slow and memoryless, and hence decorrelated. That is, neither a pure downhill nor a pure two-state folding protein domain can operate as an effective core oscillator. However, we do find that the oscillator can be remarkably efficient if it can interconvert between downhill and two-state scenarios in response to subtle changes in environment at an inherently high rate. We contend that such alternating downhill vs two-state interconversions might constitute an important natural design principle to attain optimal coordination of multistep processes without the need for additional energy inputs. That mechanism is consistent with what is known of how transcription factors perform their genome target searches. We further note that the oscillatory machine model outlined in this work makes predictions that are in close agreement with the experimental turnover rate of the multienzymatic pyruvate dehydrogenase complex, a paradigm of oscillatory behavior. Our observations support the significance of the downhill vs two-state conversion mechanism and its usefulness as tool for predicting or designing oscillatory machines.