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Globalization has given a powerful impetus to the development of international commercial activity and logistics management systems taking full advantage of cross-border networking. The solution lies at the intersection of information technologies, technical means of machine-to-machine (M2M) interaction, mobile high-speed networks, geolocation, cloud services, and a number of international standards. The current trend towards creating digital logistics platforms has set a number of serious challenges for developers. The most important requirement is the condition of sustainability of the obtained solutions with respect to disturbances in the conditions of logistics activities caused not only by market uncertainty but also by a whole set of unfavorable factors accompanying the transportation process. Within the framework of the presented research, the problem of obtaining the conditions for the stability of solutions obtained on the basis of mathematical models is set. At the same time, the processes of transferring not only discrete but also continuous material flows through complex structured networks are taken into account. This study contains the results of the analysis of the stability of solutions of differential systems of various types that simulate the transfer processes in network media. Initial boundary value problems for evolutionary equations and differential-difference systems are relevant in logistics, both for the discrete transportation of a wide range of goods and for the quasi-continuous transportation of, for example, liquid hydrocarbons. The criterion for the work of a logistics operator is the integral functional. For the mathematical description of the transport process of continuous and discrete media, a wide class of integrable functions are used, which adequately describe the transport of media with a complex internal rheological structure.