The notion personalization lies on the core of a real-world product search system, whose aim is to understand the user's search intent in a fine-grained level. The existing solutions mainly achieve this purpose through a coarse-grained semantic matching in terms of the query and item's description or the collective click correlations. Besides the issued query, the historical search behaviors of a user would cover lots of her personalized interests, which is a promising avenue to alleviate the semantic gap between users, items and queries. However, as to a specific domain, a user's search behaviors are generally sparse or even unavailable (i.e., cold-start users). How to exploit the search behaviors from the other relevant domain and enable effective fine-grained intent understanding remains largely unexplored for product search. Moreover, the semantic gap could be further aggravated since the properties of an item could evolve over time (e.g., the price adjustment for a mobile phone or the business plan update for a financial item), which is also mainly overlooked by the existing solutions. To this end, we are interested in bridging the semantic gap via a marriage between cross-domain transfer learning and knowledge graph. Specifically, we propose a simple yet effective knowledge graph based information propagation framework for cross-domain product search (named KIPS). In KIPS, we firstly utilize a shared knowledge graph relevant to both source and target domains as a semantic backbone to facilitate the information propagation across domains. Then, we build individual collaborative knowledge graphs to model both long-term interests/characteristics and short-term interests/characteristics of a user/item respectively. In order to harness cross-domain interest correlations, two unsupervised strategies to guide the interest learning and alignment are introduced: maximum mean discrepancy (MMD) and kg-aware contrastive learning. In detail, the MMD is utilized to support a coarse-grained domain alignment over the user's long-term interests across two domains. Then, the kg-aware contrastive learning process conducts a fine-grained interest alignment based on the shared knowledge graph. Experiments over two real-world large-scale datasets demonstrate the effectiveness of KIPS over a series of strong baselines. Our online A/B test also shows substantial performance gain on multiple metrics. Currently, KIPS has been deployed in AliPay for financial product search. Both the code implementation and the two datasets used for evaluation will be released online publicly(1).