Do older models always lag behind/run slow?

The processor in the iPhone 5S (the same processor in the latest iPad models) is almost 4X faster than that in the iPhone 4S. iOS7 is made to be compatible with the 4S and should run smoothly, but the hardware in the phone will limit its performance speed when compared to the 5 and in particular the 5S.

Older products should run just as well on newer operating systems, sometimes even better than they ran on previous operating systems.

Yes, it will run smoothly. The new processor (A7X) is super fast and the first 64 bit processor used in any iOS device.

When your 4S drags, the first thing to do would be to close background applications. This always help a subtle lag.

iOS: Force an app to close - Support - Apple

If You Think iOS 7 Feels Slow Here’s How to Speed It Up

http://osxdaily.com/2013/09/23/ios-7-slow-speed-it-up/http://osxdaily.com/2013/09/23/ios-7-slow-speed-it-up/

You may have many apps open which can possibly cause the slowdown and possibly the loss of wifi. In iOS 4-6 double tap your Home button & at the bottom of the screen you will see the icons of all open apps. Close those you are not using by pressing on an icon until all icons wiggle - then tap the minus sign. For iOS 7 users, there’s an easy way to see which apps are open in order to close them. By double-tapping the home button on your iPhone or iPad, the new multitasking feature in iOS 7 shows full page previews of all your open apps. Simply scroll horizontally to see all your apps, and close the apps with a simple flick towards the top of the screen.

 Cheers, Tom 😉