To foster academic integrity in students and future scholars, it is essential to understand how their integrity behaviours evolve throughout their educational trajectory and across various academic integrity topics. While much research has examined students' perception of and engagement in plagiarism and other forms of clear-cut cheating, grey-zone practices have largely been neglected, and comparisons across educational levels are rare. This paper presents a comprehensive overview of European students' conceptions of and engagement with less clearcut aspects of academic integrity, and the potential effects of academic integrity training. The study draws on a large-scale survey of 3,297 students from Denmark, Ireland, Portugal, and Switzerland, covering three educational levels (upper secondary, Bachelor, and PhD). The survey examined perceptions of and engagement in likely grey-zone and non-compliant practices across three dimensions of academic integrity: i) Plagiarism and citation practice, ii) Collaborative practices, and iii) Data collection and analysis. Responses were analysed using descriptive statistics and regression analyses. Results showed that participants at higher educational levels were better at identifying likely non-compliant practices related to plagiarism and citation, and they were less likely to have engaged in such practices during their current studies. Progress along the educational trajectory was less pronounced regarding collaborative practices and practices related to data collection and analysis. In particular, 14% of the PhD level participants admitted having deleted deviating data "based on a gut feeling that they were inaccurate" and 20% admitted to keeping inaccurate records. All participants had a low level of competence in identifying grey-zone practices, and strikingly, their competences did not improve along their educational trajectory. Academic integrity training was not consistently correlated with any group of participants' competences regarding likely grey-zone practices, although it was positively correlated with upper secondary and PhD participants' competences concerning certain likely non-compliant practices. These results call for a different approach to academic integrity training. In particular, they call for more comprehensive approaches that include grey-zone as well as non-compliant practices, and address a broad range of questionable behaviours, not only plagiarism.