In the protonation of an enolate ion, the kinetic product is the enol and the thermodynamic product is a ketone or aldehyde. Carbonyl compounds and their enols interchange rapidly by proton transfers catalyzed by acids or bases, even in trace amounts, in this case mediated by the enolate or the proton source.
In the deprotonation of an unsymmetrical ketone, the kinetic product is the enolate resulting from removal of the most accessible α-H while the thermodynamic product has the more highly substituted enolate moiety. Use of low temperatures and sterically demanding bases increases the kinetic selectivity. Here, the difference in pK between the base and the enolate is so large that the reaction is essentially irreversible, so the equilibration leading to the thermodynamic product is likely a proton exchange occurring during the addition between the kinetic enolate and as-yet-unreacted ketone. An inverse addition (adding ketone to the base) with rapid mixing would minimize this. The position of the equilibrium will depend on the countercation and solvent.
If a much weaker base is used, the deprotonation will be incomplete, and there will be an equilibrium between reactants and products. Thermodynamic control is obtained, however the reaction remains incomplete unless the product enolate is trapped, as in the example below. Since H transfers are very fast, the trapping reaction being slower, the ratio of trapped products largely mirrors the deprotonation equilibrium.