What migration policies do people in receiving countries prefer, how does political trust moderate these preferences, and for whom does this matter? Governments rely on a degree of public trust as they develop and implement policies in contentious domains such as migration. Existing studies from the United States and Europe suggest that political trust is positively related to migration attitudes and preferences. Yet the evidence base for this relationship displays several limitations: it tends to view migration policy as being narrowly about overall levels rather than covering multiple dimensions; it ignores how political trust may selectively matter for some areas involving migrants than others; and it does not systematically explore how preferences vary among population segments either in terms of overall importance or the moderating impact of political trust. In response, we report findings from two conjoint survey experiments fielded in Colombia (N=2,508) and Peru (N=2,538) that measured public support for migration policies comprising several elements available to legislators. These countries have received the most displaced Venezuelans as part of one of the world's largest migratory flows, and also display some of the lowest levels of political trust globally. Our results reveal that respondents hold multidimensional preferences as well as attach greater importance to specific policy areas (i.e. migrants' labor market access, family reunification conditions, and caps on inflows). We also show how levels of political trust selectively matter for some dimensions (i.e. employment and numerical limits) but not others. Then, using latent class analysis (LCA), we identify how these preferences organize around distinct population segments corresponding to open, conditional, and closed sets of policies. Our study makes three contributions to understanding migration policy preferences. First, it extends empirical examination of preferences--largely deriving from the experiences of high-income destination countries--to low- and middle-income settings that actually host larger shares of migrants, and which have different policy options and concerns. Second, it adds greater detail to evidence about how people distinguish among policy options, how much weight they place on those options, and for whom those options matter more. Third, it highlights the consequences of political trust for policy preferences, which has implications for understanding government functioning in low-trust environments.