This paper investigates the performance of non-orthogonal multiple access (NOMA)-based coordinated direct and relay transmission (CDRT) networks with short-packet communications undergoing Nakagami- fading channels and subject to imperfect successive interference cancellation (SIC). To enhance the transmission reliability of a cell-edge user (CE) or a cell-center user (CU), a dynamic relay selection (RS) strategy is proposed, in which exact closed-form expressions are derived for the block-error-rate (BLER) of the users and system throughput. To obtain further insights, an asymptotic BLER analysis is carried out at high signal-to-noise ratio (SNR) regime, based on which the diversity orders are deduced. Numerical results are presented to corroborate the analytical results, and they show that: (i) the performance of the NOMA system outperforms its orthogonal multiple access (OMA) counterpart in terms of BLER and throughput; (ii) the BLERs of the users are improved significantly by employing the proposed RS strategy compared with benchmark schemes; (iii) CU's diversity order is primarily determined by the fading parameters, whereas the diversity order of CE relies on both fading parameters and number of relays; and (iv) the proposed system can achieve maximum throughput performance by handling a reasonable channel coding ratio.