Cross-Domain Sequential Recommendation (CDSR) has recently gained attention for countering data sparsity by transferring knowledge across domains. A common approach merges domain-specific sequences into cross-domain sequences, serving as bridges to connect domains. One key challenge is to correctly extract the shared knowledge among these sequences and appropriately transfer it. Most existing works directly transfer unfiltered cross-domain knowledge rather than extracting domain-invariant components and adaptively integrating them into domain-specific modelings. Another challenge lies in aligning the domain-specific and cross-domain sequences. Existing methods align these sequences based on timestamps, but this approach can cause prediction mismatches when the current tokens and their targets belong to different domains. In such cases, the domain-specific knowledge carried by the current tokens may degrade performance. To address these challenges, we propose the A-B-Cross-to-Invariant Learning Recommender (ABXI). Specifically, leveraging LoRA's effectiveness for efficient adaptation, ABXI incorporates two types of LoRAs to facilitate knowledge adaptation. First, all sequences are processed through a shared encoder that employs a domain LoRA for each sequence, thereby preserving unique domain characteristics. Next, we introduce an invariant projector that extracts domain-invariant interests from cross-domain representations, utilizing an invariant LoRA to adapt these interests into modeling each specific domain. Besides, to avoid prediction mismatches, all domain-specific sequences are aligned to match the domains of the cross-domain ground truths. Experimental results on three datasets demonstrate that our approach outperforms other CDSR counterparts by a large margin. The codes are available in https://github.com/DiMarzioBian/ABXI.