Compositions including a thermoplastic elastomer or thermoplastic vulcanizate derived from a styrenic block copolymer having a reactive or crosslinkable hard block including aromatic vinyl repeat units and also preferably a crosslinkable soft block, and a non-olefin thermoplastic polymer or copolymer and preferably a compatibilizer such as the reaction product of a non-olefin thermoplastic polymer and a functionalized polymer such as a maleic anhydride functionalized styrenic block copolymer that is compatible with the styrenic block copolymer having a reactive or crosslinkable hard block, and optionally a linking compound. The vulcanizates are prepared by crosslinking the styrenic block copolymer in the presence of the non-olefin thermoplastic polymer and a suitable crosslinking agent, and optionally the compatibilizer, preferably utilizing dynamic vulcanization. In a further embodiment, thermoplastic vulcanizate compositions are provided including the styrenic block copolymer, a polyolefin polymer or copolymer, a non-olefin thermoplastic polymer and the compatibilizer.