The concept's weirdly popular right now, right? It usually follows a pattern where the assassin's cold efficiency gets repurposed. A lot of stories have them start by sizing up their new family and kingdom as a new operational environment. They don't get emotional, they do threat assessments.
I like it when the narrative leans into the dissonance. A toddler speaking with the cadence of a seasoned killer, or using their old-world knowledge of poisons to 'help' in the kitchen. The adaptation isn't about becoming a good person, but about applying a lethal skill set to new problems, like noble politics or dungeon crawling. The old life is a tool, not a burden to overcome.
My pet peeve is when the story forgets the 'assassin' part halfway through and they just become another overpowered isekai protagonist. The best ones make their past haunt their choices in subtle ways, like an inability to trust or a reflex to calculate escape routes in every room.