Graphynes, a class of two-dimensional carbon allotropes, exhibit exceptional electronic properties, similar to graphene, but with intrinsic band gaps, making them promising for semiconducting applications. The incorporation of acetylene linkages allows for systematic modulation of their properties. However, the theoretical characterization of graphynes remains computationally demanding, particularly for electron-phonon coupling (EPC) analyses. Here, we employ the density functional tight binding method within the DFTBEPHY framework, providing an efficient and accurate approach for computing EPC and transport properties. We investigate the structural, mechanical, electronic, and transport properties of graphynes, comparing transport calculations using the constant relaxation-time approximation and the self-energy relaxation-time approximation (SERTA) alongside analytical models based on parabolic- and Kane-band approximations. For graphyne, the SERTA relaxation time is 0.63 (1.69) ps for holes (electrons). In graphdiyne, the relaxation time is 0.04 (0.14) ps for holes (electrons). While the hole mobilities in graphyne are on the order of 103 cm2 V-1 s-1, the electron mobilities reach up to 104 cm2 V-1 s-1. In graphdiyne, the mobility values for both types of charge carriers are on the order of 102 cm2 V-1 s-1. The phonon-limited mobilities at room temperature in graphyne fall between those of graphene and MoS2, while in graphdiyne, they are comparable to those of MoS2.