Neural Architecture Search (NAS) has emerged as a crucial method for automating the design of deep learning models. Despite its potential, NAS frequently requires substantial computational and hardware resources. To mitigate these challenges, transferable NAS (TNAS) has been introduced, leveraging prior NAS results to enhance performance on new tasks. However, existing methods largely focus on knowledge transfer within identical neural search spaces, overlooking the potential for cross-domain transferability. Motivated by this gap, we explore evolutionary TNAS across heterogeneous search spaces by learning common neural representations. In particular, we introduce a novel approach that encodes both operational and topological information of neural architectures into a unified sequence using a simple tokenizer. This sequence is then processed by a variational auto-encoder, with a Transformer-based encoder to capture rich neural representations and a decoder that reconstructs the original sequence. By utilizing these latent representations, we further establish an inter-domain mapping that acts as a bridge, enabling effective explicit solution transfer among diverse search spaces to enhance the evolutionary NAS process. To harness this capability, we develop an evolutionary sequential transfer optimization approach that transfers knowledge during population initialization, providing both flexibility and adaptability. To the best of our knowledge, this work serves as the first attempt in the literature exploring evolutionary TNAS across diverse spaces. Moreover, we demonstrate the utility of our method through comprehensive empirical studies using different architecture spaces, including NAS-Bench-101, NAS-Bench-201, and the DARTS search space. Our results show that the proposed method significantly enhances the adaptability and performance of NAS across varied domains.