Professional cinemas license films directly from the major Hollywood studios. The studios, as owners of the intellectual property, directly control which licenses they make available for which of their films. The terms of these licenses are negotiated on a per-studio, per-theater, and per-title basis, but usually they offer a flat rate of between $250 and $350 or a minimum of $250 vs 35% of the box office.
However, there are some practical distribution limitations. As you are probably aware, the industry switched over exclusively to using digital projection technology about five years ago. The vast majority of cinemas no longer have the ability to play film prints. Further, maintenance of old film prints is expensive, as is converting them to the high-quality digital print required for cinema projection.
Thus, giving these limitations, the major studios have a designated set of "repertory" titles which they make available for digital cinemas. The content cinemas can play is generally limited to these titles. Other studios, particularly the "newer" variety (e.g. Amazon Studios) will allow content to be played via their streaming service or Blu Ray disk. However, the quality of these secondary sources is significantly worse and, in my opinion, unsuitable for projection in a cinema.
The only studio that regularly does not offer repertory titles is Walt Disney Studios Motion Pictures. Their business practice has been to have exclusive windows for theatrical, on-demand, and purchased content, and cinema repertory is not something they offer. And, as someone mentioned, I don't know what will happen with the Fox properties - Fox has traditionally been one of the biggest repertory suppliers. If I had to guess, I'd say that Disney leaves Fox and their business model more or less intact for now.
Regarding other services (Swank, etc.) - these license arrangements specifically exclude exhibition in a professional cinema. If a cinema were to procure a license through such a service, it would likely be void as theaters have master licensing agreements with studios which would override any third party arrangements.