Let sets of functions Z and Л on the time interval T be given, let there also be a multifunction (m/f) a acting from П to Z and a finite set Д of moments from T. The work deals with two questions: the first one is the connection between the possibility of stepwise construction (specified by Д) of a value z of a(w) for an unknown step-by-step implemented argument u> e and the existence of a multiselector /3 of the m/f a with a non-antieipatory property of special kind defined by Д; and the second question is how to build the above /3 for a given pair (а, Д). The consideration of these questions is motivated by the presence of similar step-by-step procedures in the differential game theory, for example, in the alternating integral method, in pursuit-evasion problems posed with use of counter-strategies, and in the method of guide control. It is shown that the step-by-step construction of the value z € a(u>) can be carried out for any in steps implemented argument ui if and only if the multiselector /3 is non-empty-valued. In this case, the desired value z can be selected from Р(ш) in step-by-step procedure for any unknown in advance argument u>. The key point of the work is the procedure for calculation the multiselector /3, for which a constructive and finite-step description is given. Illustrative examples are considered that include, in particular, problems of a guaranteed result optimization under functional constraints on control and/or disturbance implementations.