Hey just did this today don't know how useful it will be to all of you.
When running random dungeons with someone from your server after you finish the first dungeon and people drop out if you keep two people in group you can run to another dungeon of your choice and have the leader click on the look for more option and you will start the dungeon that you are zoned into. You get the Random Group buff (5% wtv) but I'm not sure if you get the random triumphs at the end I forgot to check I'll try it again tomorrow and report back in.
sum up:
After running a random dungeon and you and someone else from your server are left in group (someone has green leader flag)
run to instance of your choice and click the option to find members to replace the ones that have dropped out.
counts as random heroic