Effect of alloy structure of Mn-rich Ni$_{52.6}$Mn$_{34.5}$Sn$_{12.9}$ and Ni$_{52.9}$Mn$_{33.8}$Sn$_{13.2}$ Heusler alloys on their magnetic and transport properties is investigated. As shown, the as-cast alloys have two-phase structure. Annealing at T = 1273 K and following quenching of these alloys cause the formation of well-ordered $L2_{1}$-type single-phase structure. Both as-cast alloys demonstrate anhysteretic martensitic transformation within the temperature range 300 K < T < 500 K, which is accompanied with significant changes in their magnetic and transport properties. Based on noticeable changes of magnetic and transport properties of annealed Ni$_{52.6}$Mn$_{34.5}$Sn$_{12.9}$ alloy, it is concluded that, within the temperature range 300 K < T < 400 K, two martensitic transformations in this alloy take place: austenite—10Mmartensite (290K < T < 390 K) and 10М martensite—martensite (110 K < T < 210 K).