For the research, I suggest a personal and group interview. For authenticity, I also recommend it in communities that live in a place near the border where migrants cross the border. The objectivity and authenticity of the questionnaire and the online social media interfaces are very objectionable. I also recommend databases of authentic national and international organizations and referenced sources of thematic research for background information.
The emitting, transit and receiving areas, as well as the direction and magnitude of migration must be taken into account. Given the migration pattern, I suggest clarifying the methodology and resources.
The type of data will depend on the study, whether it is qualitative or quantitative. If you are after quantitative data, then some large scale surveys will provide basic information, such as country of birth, country of residence a year before the survey, nationality. In the EU, Labour Force Surveys provide such information. The caveat with all surveys is that they often neglect some sub-populations e.g. those residing in institutional households and the sample sizes may not be large depending on what exactly you need to measure. They also come with sampling and other errors.
If you're looking for data collected specifically on migrants, then one good example is https://mafeproject.site.ined.fr/en/ Migrations between Africa and Europe (MAFE) project's data.
In the UK, there is the International Passenger Survey that has information about travellers and those who plan to stay in a country for more than 12 months (i.e. migrants).
Other data with much better coverage are census microdata. However, the analysis of them will be restricted and you may need special permissions and trainings to do so, plus access of the data is often via secure environments. An example from the UK Data Service: https://census.ukdataservice.ac.uk/get-data/microdata.aspx.