Images should stay put!
I alternate between using Word for desktop and Word online. If I make a document on my desktop version and insert an pictures, the pictures mess everything up if I pull it onto Word Online. The captions and images move all over the place and trying to fix them just makes it worse. It would be great if this problem was resolved.
I have found that right-clicking on pictures, then selecting "Wrap Text," and then selecting "Square," "Tight," "Top and Bottom," or "In Front of Text," makes pictures easier to deal with. They can be moved with greater precision and are more likely to stay where you anchor them on the page. I have not tried switching between the desktop and online versions, but I have done so across computers running different versions of desktop Word. It used to be a nightmare of pictures all over the place until I learned this trick. I hope it helps!