Migrants often face a complex situation for navigating services and administrative processes to transition and integrate into a new environment. They rely on guidance from service advisors and informal peers like other migrants to assist them. Service advisors try to understand, explore, and resolve migrant concerns through conversational interaction, while often seeking advice from other domain experts. The emergence of artificial intelligence (AI)-mediated communication holds the potential for enhancing communication between migrants and service advisors while leveraging the expertise of service advisors and facilitating migrants' autonomy in the integration process. Our research investigates the social context, mental models, and challenges encountered in guidance counseling for migrants in Finland, while examining what role AI-mediated communication can play in supporting the process. Based on our findings, we propose design implications for improving shared understanding between service advisors and migrants through salient summarization of conversations and creating collaborative visual narratives. Our findings also indicate the crucial role of leveraging peer-based knowledge between (and within) service advisors, domain experts, and migrants, which can be facilitated by AI-based tools. Our exploratory research offers insights into human-centered AI integration in the migration process, as well as the opportunities and challenges of using AI-mediated communication to improve the migrant experience.