We present a theory of a conformational collapse-to-swelling transition that occurs in aqueous dispersions of multiresponsive (pH- and thermoresponsive) microgels upon variation of ionic strength, temperature, or pH. Our theory is based on osmotic balance arguments and explicitly accounts for ionization equilibrium inside microgel partices. The theory predicts complex patterns in the dependence of the microgel particle dimensions on the control parameters: An increase in temperature leads to worsening of the solvent quality for the gel forming LCST-polymers and to concomitant decrease in the dimensions of the gel particles. This collapse of the gel particles provoked by an increase in temperature occurs either smoothly (at high or low ionic strength), or may exhibit a jump-wise character at intermediate ionic strength. The theory further predicts that the degree of swelling of microgel particles varies nonmonotonously and exhibits a maximum as a function of salt concentration at a pH close to the pK. This nonmonotonous variation of the particle dimensions occurs continuously at temperatures below or slightly above LCST (good or marginal poor solvent strength conditions, respectively), whereas at higher temperatures the jump-wise swelling of the gel particles is followed by either continuous or jump-wise collapse induced by progressive increase in the salt concentration. A decrease/increase in pH leads to deswelling of the weak polyacid/polybase gel particles, which occurs smoothly at temperatures below LCST, but may exhibit a discontinuity above LCST. These theoretical predictions can be used for design of smart stimuli-responsive microgels.