I have done it before and I would definitely recommend it, but you probably should try to work at least one closing shift every week or at least a late mid to see your closing team once in awhile.
Also you should level down one day and do the closing duties yourself so you can see how much work should be getting done on average. (They will know you mean business and maybe even respect you more)
I would create a checklist of their responsibilities and have them check them off, sign it, and turn it in to your mailbox every night. If they don't complete all of them, tell them to write a detailed description on why not. I'm recommending this, since you said your LODs are not following up enough. Now if they continue to not meet expectations, then simply coach them and tell them it's not acceptable. And by you working closing shifts for a week and maybe once a week, they can't give you any excuses because you know it can be done.
My guess is your LODs don't follow up not because they don't care but because they don't know enough of your particular work center too follow up. I would spend some time with the ones that aren't well experienced with logistics and show them what to look out for and tell them how much you would appreciate their follow up. What I do is tell the closing team member to always check out with the LOD before they leave and the LOD leaves it in their closing email. (Writing a Vibe card or a thank you email for when they do follow up always helps)