Can one put photos in plastic frames that have either swiveling plastic latches or soft metal bending latches that open the back and let a photo be placed between the glass and the back without glue?
I think it depends on the type of photo and the type of frame. A rare/previous/important picture or a delicate frame would fall in the halachic category of mukze chisaron kis ("a loss of the pocket") and could not be moved.
This category of muktze relates to objects that are valuable to their owner (regardless of their monetary value), that don't have a specific/permitted use on Shabbat and that people are particular not to employ for alternative uses (see e.g., Mishna Brura 308:1). Expensive and/or fragile objects are almost certainly included in that category. See R Daniel Braude's Learn Shabbos, pp. 455ff.
R Dovid Ribiat (the 39 melochos, vol. 1, p. 47) actually explicitly mentions delicate family photographs as examples of mukze chisaron kis.
If the frame is a cheap plastic frame and the picture unimportant, they do not fall into this category and can be handled. There is no issue of bone or make be patish as you are not building something new when opening/closing the frame (like a folding chair).