Is it possible to find an unpaid and "half-time" postdoc? I've defended my thesis, and I would like to do my postdoc abroad. But the problem is that I am married, and I have two little children. So, I can't just leave my family for a couple of years, and neither I can bring them with me (too expensive, too difficult etc.). 

I have a couple of colleagues in the same situation, and they seem to find a solution: a kind of unpaid postdoc (they receive only small sums of money covering travel expenses) which supposes visiting the inviting university regularly but without staying there for a long time. In this format they prepare a monograph and give some seminars, it is counted as a postdoc by ANECA (homologation institution) here in Spain but doesn't require full-time occupation if you can maintain certain level and volume of academic publications. In both cases they were invited unofficially by their postdoc coordinators, it was not an announced grant or scholarship.  

For me it could be the best option. It will give me an opportunity to get some foreign postdoc experience, stay on academic track (in Spain it is really difficult in the umanities) and at the same time I don't have to leave or move my family...   

But I really don't know how to start. Is there any opportunity to get an information which universities offer this way of collaboration and how to apply? In the case of one of my colleagues, it is one of the Swiss universities and in another case it's Finland, but the information offered by these universities on their webpages is related only to the "normal," fellowship-based, full-time postdocs. And maybe there are some other opportunities of this kind in other countries and universities?
I know that some institutions can offer the "associated member" status to a researcher who is not on their payroll. The researcher needs to be recommended by a senior professor from the institution, who typically presents a project with this researcher to justify the request. However I've seen it used only with somebody after they had been a postdoc in the institution. 

Obviously one needs at least to prepare a research project with a member of the institution, so I think the best way would be to contact the academics in your network who could be interested in a project with you. I'd suggest to propose to work with them for free on a grant proposal, since this would be interesting for both the PI and for the institution. If you find a suitable EU call, you could even try to involve a partner in Spain which would hire you if the grant is successful.