We present a unified asymptotic framework for transmit antenna selection in multiple-input multiple-output (MIMO) multirelay networks with Rician, Nakagami-m, Weibull, and generalized-K fading channels. We apply this framework to derive new closed-form expressions for the outage probability and symbol error rate (SER) of amplify-and-forward (AF) relaying in MIMO multirelay networks with two distinct protocols: 1) transmit antenna selection with receiver maximal-ratio combining (TAS/MRC) and 2) transmit antenna selection with receiver selection combining (TAS/SC). Based on these expressions, the diversity order and the array gain with M-ary phase-shift keying and M-ary quadrature-amplitude modulation are derived. We corroborate that the diversity order only depends on the fading distribution and the number of diversity branches, whereas the array gain depends on the fading distribution, the modulation format, the number of diversity branches, and the average per-hop signal-to-noise ratios (SNRs). We highlight that the diversity order of TAS/MRC is the same as TAS/SC, regardless of the underlying fading distribution. As such, we explicitly characterize the SNR gap between TAS/MRC and TAS/SC as the ratio of their respective array gains. An interesting observation is reached that for equal per-hop SNRs, the SNR gap between the two protocols is independent of the number of relays.