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That lag is likely caused by background sync processes that were not running while the phone was shut off. Processes such as mail caching, photos thumbnails, Play Services updates, scanning for app updates, plus data updates for apps such as Facebook.
How often do you shut it off and for how long?
I still have my 2013 Moto X connected to my Google account, I experience this very thing when I boot it every so often and the items listed above are the exact reasons why it lags for a little while after I boot it, based on the barrage of notifications and updates that come through during that time.