This is usually done in equilibrium or kinetic studies in order to reduce relative changes in the ionic strength of a solution.
The real goal is to reduce changes in the activity coefficients of ionic species which allows the definition of conditional equilibrium or rate constants.
They are supposed to only influence the ionic strength.
Typical inert salts that are used include: NaClO4, NaCl, KNO3, NaNO3, triflates (e.g. NaOSO2CF3[1]).
Inert salts are never perfectly inert and their use will always interfere with the process under investigation, although the influence may be negligible.