Most people don't know that taking to the Apple Store, they will ask you set of questions.
1) Did you restore your backup or set it up as a new phone?
2) If you restored it from a backup, they will tell you that there might've been some old corruption from the older phone you have had that carried over. There's a 50/50 debates on this issue itself. Some say there's no way to carry over corrupted files, and some say you cant.
(I've worked for Apple for 2 years in the past as a tech and corporate level, I can definitely tell you, you CAN carry it over to your new device. Regardless what people are saying, restoring actually isn't the best way for new phones believe it or not.)
3) They will advise you to set up your new phone as a NEW phone without any backups. You have the option to do it in the store or do it at home.
I recommend everyone setting up your new phones as NEW. Don't restore from backup via iCloud or iTunes. We can always debate back and forth about corruption coming back and forth, but I've been doing this ever since the first iPhone came out. I don't want to debate anyone here haha, all I want is to help and assist people. So if you're having issues, do the following:
1) SET UP YOUR PHONE AS NEW (NOT BACKUP)
2) TEST OUT THE LAG AND ISSUES YOU ONCE HAD
3) IF IT CONTINUES, THEN GO BACK TO APPLE AND LET THEM KNOW YOU DID A SET UP AS NEW
I tested this myself on my new iPhone 6s Plus just to see if there was issues, and lo and behold, there were issues from the backup. I set it up as new and everything works as designed flawlessly.