The invention provides compositions comprising at least one propylene-based polymer, at least one peroxide and at least one coagent. In a preferred example, a composition comprising at least one propylene-based polymer, at least one peroxide, at least one grafting agent, and at least one coagent, and wherein the at least one coagent has a substituted carbon-carbon double bond structure selected from the group consisting of structures (I), (II), (III), (IV) and (V) as follows, including cis and trans isomers wherein D1, D2, D3 and D4 are each, independently, one of the following: (a) a hydrocarbon substituent containing a heteroatom with a nonbonding pair of electrons, and where the heteroatom with the nonbonding pair is directly attached to the carbon-carbon double bond, or (b) a hydrocarbon substituent containing a carbon-carbon pi bonding pair, and where the first carbon of the carbon-carbon pi bonding pair is directly bonded to the carbon-carbon double bond; and wherein the R1, R2 and R3 are each, independently, H, a hydrocarbon substituent, or a substituted hydrocarbon substituent; and wherein R1, R2 and R3, each, independently, does not have the same chemical structure as D1, D2, D3 or D4; and wherein the molar ratio of the “coagent(s)” to “free radicals formed from the peroxide(s)” is from 0.5:1 to 5:1; and wherein the coagent selected from structures (I)-(V) does not contain only one phenyl group, or does not contain only one furyl group.