This paper extends finite-field multiple-access (FFMA) techniques from binary to general $p$-ary source transmission. We introduce element-assemblage (EA) codes over GF($p^m$), which generalize element-pair (EP) codes, and define two specific types for ternary transmission: orthogonal EA codes and double codeword EA (D-CWEA) codes. We propose a unique sum-pattern mapping (USPM) constraint for the design of uniquely-decodable CWEA (UD-CWEA) codes, which include additive inverse D-CWEA (AI-D-CWEA) and basis decomposition D-CWEA (BD-D-CWEA) codes. Additionally, we introduce non-orthogonal CWEA (NO-CWEA) codes and their corresponding USPM constraint in the complex field. Furthermore, $p$-ary CWEA codes are constructed using a basis decomposition method, leveraging ternary decomposition for faster convergence and simplified encoder/decoder design. We present a performance analysis of the proposed FFMA system from two complementary perspectives: channel capacity and error performance. We demonstrate that equal power allocation achieves the theoretical channel capacity, and then investigate the finite blocklength (FBL) characteristics of FFMA systems. Moreover, we develop a rate-driven capacity alignment (CA) theorem based on the capacity-to-rate ratio (CRR) metric for error performance analysis. Finally, we compare $p$-ary transmission systems with classical binary transmission systems, revealing that low-order $p$-ary systems (e.g., $p = 3$) outperform binary systems at small loading factors, while higher-order systems (e.g., $p = 257$) excel at larger loading factors. These findings highlight the potential of $p$-ary systems, although practical implementations may benefit from decomposing $p$-ary systems into ternary systems to manage complexity.