Die Familie hat auch einen kurzen Text unter die Bilder geschrieben.
Based upon the discussion in the post "Why does 'unter' not take dative case in 'unter etwas fallen'”? there should be movement in a direction, but is writing text movement in a direction?
RDBury explained it well, I'll just add that this is true for most of the verbs that have to do with writing, painting, drawing, scribbling, chiselling etc. They all take prepositions of direction answering the question "Wohin?" for the place where something is written/painted/etc.
So that's kind of a rule in itself.
To write, paint, draw something somewhere is seen as akin to putting something somewhere where it wasn't before. Or, if you will, the paint or ink is what is moved.
Unser Sohn hat gestern sein Tag auf die Garage des Nachbarn gesprayt.
The movement here is figurative. Think of the family as carrying the text around, perhaps in their minds or in the ink they used. Then they left it under the pictures by writing it there. Or perhaps it would be better to rephrase the rule to say you use the accusative when there is movement or a change. The important thing is to not take grammatical rules too literally. Natural language seems to be inherently fuzzy and there will always be edge cases where you have to apply a certain amount of intuition to get the correct phrasing.
It depends whether you can ask 'wo' or 'wohin', thus whether a place is referred to or direction or movement.
Unter dem Bild schrieb er --> the image was held above him while he wrote (whatever)
Unter das Bild schrieb er --> he wrote something onto a place below the image
One can even combine it of course:
Unter dem Bild schrieb er etwas unter das Bild. --> he wrote something under the image while being below the image.
This sense of direction is not always very clear, and you can think of the text being moved from a mental place to the position under the image. Other examples might make it slightly clearer:
Wir gehen unter der Brücke --> we are walking under the bridge (but already are there)
Wir gehen unter die Brücke --> we (are going to) walk under that bridge (over there), thus need to still get there
Wir sehen unter dem Baum eine Eule --> we see a owl under that tree (the owl is under the tree and we see her with her place fixed)
Wir setzen uns unter den Baum zum Picknick --> We will (go and) sit under that tree for a picknick. Thus a sense of direction or movement is implied.
I think the dative case would imply that the family is staying under the images when they are writing a text, which text might be in a notebook, for example.
