For DV cases you will have to be in status when USCIS adjudicate your application ( that is when they make a decision).
It doesnt matter when you will file your application (mail it in). what matters is when it will be adjudicated (for DV usually on the interview date). So the main concern is whether you are in status when its is being adjudicated not when its being filed.
DONOT compare adjustment of status based on marriage or asylum or anything else to DV cases. For example adjustment of status cases based on marriage can be approved even if the applicant is out of status. BUT those are different We are here about DV and the rule is you have to be in status when the application is being adjudicated (that is when the officer ask you in the interview if you are in status right there and then).
Whats the status of your kids? Are they US citizens? or are they dependents on your L visa??
If they are US citizens then they have no problem being here regardless of your visa status( yet you do have a problem because you are not a citizen and you have to hold a valid status)
If they are dependents on your visa, then as soon as your status is terminated they are being faced with the same because their residency here depends on you. Going to school is not permitted beyond that authorized stay which ends with your visa.
Unless you can extend your visa or switch to a different visa type then i think you will have to depart with your family and go for CP because AOS will not work without you and them being in full compliance with USCIS.
So to answer your Title question (Out of status after filing i485 - What happens ?)
It will simply be rejected or denied. After breaking the rules you dont get any privileges. Remember that DV is a privilege and not a right.
My CN is 2014AS000019xx . Currently on L1 expiring on 31st Oct. Most likely I would be able to file my i485 in October when I am in Status. My question is if my L1 expires on 1st Nov 13 and if my interview is scheduled in Mar 14 (based on Boston processing time) , will i be disqualified from the DV?
My kids are in school here and I opted for AOS. My L1 cant be extended and I am hoping to get H1B1 (for Singapore) visa if i am going to be disqualified.
Thanks for advice