Markov branching systems form a fundamental class of stochastic models that are extensively applied in biology, physics, finance, and other domains. These systems are distinguished by their continuous-time evolution and inherent branching structure, allowing transitions to multiple states from a single one. This branching mechanism plays a critical role in modeling phenomena such as population dynamics, epidemic spread, and probabilistic systems with multiple outcomes. Unlike standard Markov processes, branching systems require a simultaneous treatment of transition dynamics and branching probabilities, resulting in a more intricate mathematical framework. In this work, we investigate the asymptotic properties of transition functions in continuous-time Markov branching-immigration systems. Our focus lies in refining known limit theorems, establishing convergence rates, and deriving improved asymptotic expansions under relaxed moment conditions. The results contribute to a deeper understanding of the long-term behavior and invariant structures within these systems.