Do you have a trick for getting the images as well, as opposed to them being links to the remotely hosted?

Obsidian recently introduced a native 'Download attachments for current file' which you can invoke with cmd / ctrl + p.

I like this as I don't always want all the images for something I've clipped from the web. This gives me the choice.

I think that is just a convenient way to do it with markdown.

Theoretically you could embed images in markdown with 'data:' scheme. But I am unless it is very small images it will probably not be very efficient to embedded the data in a text file.