Grant-free random access (RA) has been recognized as a promising solution to support massive connectivity due to the removal of the uplink grant request procedures. While most endeavours assume perfect synchronization among users and the base station, this paper investigates asynchronous grant-free massive RA, and develop efficient algorithms for joint user activity detection, synchronization delay detection, and channel estimation. Considering the sparsity on user activity, we formulate a sparse signal recovery problem and propose to utilize the framework of orthogonal approximate message passing (OAMP) to deal with the non-independent and identically distributed (i.i.d.) Gaussian pilot matrices caused by the synchronization delays. In particular, an OAMP-based algorithm is developed to fully harness the common sparsity among received pilot signals from multiple base station antennas. To reduce the computational complexity, we further propose a free probability AMP (FPAMP)-based algorithm, which exploits the rectangular free cumulants to make the cost-effective AMP framework compatible to general pilot matrices. Simulation results demonstrate that the two proposed algorithms outperform various baselines, and the FPAMP-based algorithm reduces 40% of the computations while maintaining comparable detection/estimation accuracy with the OAMP-based algorithm.