I just used my hand signal for down and then I used force. LOL. I took the easy way out and gently stroked the side of Lyric's head/cheek and pressed his head right down so he was lying flat and told him, "hold it..." That "hold it" he had already learned which means to not move. (luring is really better to get the head over flat) I treated and released, made a fuss. (I keep the praise calm until released so he doesn't pop up before... then more praise) Then we did it again a few times.
Next, I began to add in the cue, "bang bang!" with my hand signal and if needed would assist his head over so it was flat on the floor...."ho-o-old it....h-o-o-o-old it." Praise (low key)/treat/release. Pretty soon, if I just waited (because I'd often have to wait) he would slowly put his head all the way over onto the floor and freeze until released. Slowly, I faded the hand signal for down and turned it into more of a shooting the gun with the pointing finger signal along with the verbal cue.
The only thing we have yet to do is speed things up. He dies verrrrrrrry slowly. LOL. So, if I really cared that much about this trick, I would get out my clicker and click/treat for the faster drop to a down, head down thing. And gradually reward only for better and faster ones, not the crummy ones. LOL.
My son's dog falls over very quickly and rolls over on her back, but she doesn't freeze or lay her head flat down every time. He just did it by yelling, "BANG!" pretty loudly and forcefully. Uggg. And pointing the finger at her. (I think that's what he did. He did this before he lived here) Maybe he helped her into a down position. (?) And she's a little submissive and would roll over easily. Then he praised and treated her. I don't particularly like that idea because at first, it must have been a little overwhelming to her. She was mistreated in her childhood. But she knows now that it's a game and has fun doing it I think because she gets a good treat and loads of attention. I positively do not recommend doing it that way.
Breaking things into steps is always a good way to start out and then put them together. Very often, starting with the last step and working backwards to the front is a good way to do it....like, starting with teaching the dog to lay his head down flat on the floor, attaching a cue to that. Then teaching the down and putting the head on the floor, first as two manuvers, then putting them together and eliminating one of the two cues you may have used. Then starting from a stand to a down/head down....that sort of thing. Each part may have or develop a seperate cue or hand signal to help the dog do that part but then once the dog is doing each part easily, you can then eliminate each seperate cue (gradually) and transfer to one cue for the whole chain of behaviors.
Anyhow, you can do a Google search too for training tricks and see what there is. Good luck. Let us know how things go.