Recurrent neural networks such as Long Short-Term Memory (LSTMs) and Transformer-based Large Language Models (LLMs) are increasingly deployed in applications that rely on sequential predictions, including time-series forecasting, code generation, dialogue systems, and decision support. Ensuring their reliability, robustness, and fairness remains challenging due to the oracle problem, non-determinism, distribution shift, and rapid model evolution. Metamorphic Testing (MT) has emerged as a promising strategy to alleviate the oracle problem by specifying expected relations between multiple executions rather than absolute outputs. However, while a growing effort applies MT to LSTMs and LLMs, the landscape is fragmented across domains, targeted properties, and tooling. This paper presents a literature-driven survey of testing techniques for LSTMs and LLMs with a particular focus on MT. We first characterise LSTMs and LLMs under a unified view of sequential prediction models and summarise traditional evaluation and non-MT testing approaches (e.g., adversarial, coverage-guided, and drift-aware testing). We then review and classify existing MT-based efforts along multiple dimensions, including model type (LSTM vs. LLM), application domain, transformation patterns, targeted properties (robustness, fairness, hallucinations, regression), and degree of automation. Finally, we identify gaps and outline open challenges for designing, selecting, and operationalising MT as a unifying testing strategy for sequential prediction models.