The binding of polyelectrolyte to a temperature- and pH-responsive microgel based on poly-N-isopropylacrylamide (PNiPAM) copolymerized with methacrylic acid (MAA) as a soft and porous substrate was investigated as a function of time and temperature in order to probe rearrangements in such complexes. Oppositely charged polyelectrolytes bind to the charged microgels, and the composition of the resulting complexes stays constant with time. The number of titrable COOH groups, the size, and the electrophoretic mobility of the complexes, however, change with time due to rearrangements of polyelectrolyte chains inside of the microgel. Polyelectrolytes can be used to modify the properties of microgels. The volume phase transition temperature (VPTT) of PNiPAM-co-MAA microgels depends on the pH value, while microgel polyelectrolyte complexes collapse above the VPTT of 32 degrees C independently of the pH value. The experiments reveal that polyelectrolytes can be partially released from microgel polyelectrolyte complexes at T > VPTT. In addition, rearrangements are induced by the collapse. Rebinding of the polyelectrolyte occurs upon reswelling of the complex when the temperature is reduced below the VPTT. Such temperature cycles affect the size and electrophoretic mobility of complexes. The rearrangements can be used to increase the amount of polyelectrolyte that is bound to the microgel and are thus important for applications that rely on loading microgels with polymers. Interestingly, the colloidal stability of the complexes at T > VPTT depends on the preparation temperature; complexes prepared at T < VPTT remain colloidally stable when heated to T > VPTT; on the other hand, complexes prepared at T > VPTT display poor colloidal stability.