The past decades witness an ongoing interest in reconceptualizing writing as digital multimodal composing (DMC). The emergence of language and multimodal generative artificial intelligence (GAI) tools leads to new forms of DMC processes and products. Yet prior studies have mostly examined non-GAI-assisted DMC, and few have explored how GAI-assisted DMC processes would facilitate students' authoring of quality DMC products. This study addresses the gaps by conceptualizing the notion of quality DMC product, and comparing the composing processes and products of 14 GAI-assisted DMC groups against 10 non-GAI groups across four English classrooms. Multiple sources of data were collected for a mixed method study. The findings show that while non-GAI groups demonstrated similar performance outcomes with GAI groups in base units, layouts, and rhetorical relations, the GAI groups had significantly higher scores in the development of purpose, outline/navigation skills, and their overall DMC outcomes. A taxonomy of GAI's affordances in facilitating DMC was also identified. Based on the findings, the study develops a process-genre integrated GAI-assisted DMC model for future teachers to guide students' GAI-assisted DMC processes toward quality DMC product. This integrated model links DMC processes to products and specifies the roles of teachers and students when composing with GAI.