At the beginning of the procedure I was quite worried; my university is small (in terms of research), so that I do not have enough contact with other researchers. However, after some research on internet and speaking with some fellows from other universities I have thought of a strategy. Below I describe it through the help of a short list:
1a- you have being reading papers for your research; then write a list of the authors on which you are more interested (according to their university prestige, their and your research topics -the author you contact may be your mentor during your visiting period-, the country too -3 months or more in a place you do not like can be heavy-);
1b- look for the world ranking of the departments in your domain of research; write the list of those in which you aremore interested;
2- visit the website of the (economic) department for each of the author/university you are interested in. Check the courses they offer,the seminars, the academic cv of the other researchers of the department you are applying to, and the services they provide (e.g., very likely you won’t be provided with a desk/office);
3- prepare a list of documents: cv, phd coursework, project of thesis, a paper (if you have one, otherwise the master thesis, since it has to work simply as research skills sample)
4- write a reference letter -specific for the department and author you are contacting- for your self which will be signed by one of your referents (this one letter will be your best reference, so spend much time on it); ask to another one/two referents to write and sign a letter for you (it is important that you choose someone you trust and you give them all the important info about the department and author you are contacting). You can find on the internet samples of reference letters for phd visiting applications; you could also use reference letters that they wrote for you previously (e.g., thos that your bachelor/master’s profs wrote for you when you applied to your current phd position)
5a- carefully write the email for the first contact to the author asking for his/her mentorship; here you sum up your interests and intents, plus the salient traits of your career; as attachment, send the documents you already prepared; conlude the email by saying that upon request you can send the reference letters. Afterwards, if s/he is interested then write to the head of his/her department
5b- in case you opted for the step 1b (you have chosen the university, and so the department, not the author), write directly for the head of the department.
N.B. However, if you choose a top uni, or an author in a top uni, this list will help you only partially, since the best unis have a standardized process for the application as visiting phd student.
Be sure you know whether your uni will finance the additional expenses for your visiting. 99% of the times you won’t receive any funding from ther hosting university.
I have been organizing my visiting period in this way. I will keep you update on the results.