I'll use a boxing example. You have a short boxer fighting a tall boxer. The short boxer is constantly advancing his position. He wants to get inside and close to the taller boxer. Inside and close, the shorter boxer has the advantage, its easier for him to throw full power punches and harder for the taller boxer to hit him. If you watch a boxing match, with a short guy verses a tall guy, usually, you will see the shorter guy continuously advancing his position, trying to cut off the ring, get the taller guy on the ropes and get inside. This is similar to the grappler advancing his position. Both are moving to a position where they have an advantage and the other guy has a disadvantage.
If we consider "advancing your position" as moving to a position where you have the advantage over the other guy... Then, once our tall boxer gets pressed into the ropes... he works on advancing his position. That is he is working to get off the ropes and establish distance, where he can hit the shorter guy but the shorter guy cannot hit him. The tall guy is constantly advancing to one position, where he has an advantage and the shorter guy is constantly advancing to a different position, where he has the advantage.
Further, boxers (and all strikers) work towards angles. They want to come in at an angle, because when you come in at an angle, it gives you an advantage... its easier to land your punch, and it is harder for the other guy to defend or land his own punch. Now, you can't just step to the side and then punch... the other guy will turn and take away your angle. You have to set it up. Foot work, head movement, feints, jabs... all kinds of things are used to set up the opportunity for you to get that angle, where you can deliver your punch. Most of the time when you see two boxers dancing and feinting and circling... they are both trying to advance to a position of advantage over the other guy.
A good example of this is an orthodox boxer against a southpaw. Whoever has their lead foot to the outside, has the advantage. You will see both fighters constantly trying to establish their lead foot as the outside foot. Again they are constantly advancing their position.
Yes, in grappling you tend to maintain the dominant position for great lengths of time... you are mounted for a few minutes, you work towards establishing half guard for a few minutes and then full guard, where you hope to stay while you set up your submission or sweep to an even better position, which you then hope to keep.
Striking works in a similar manner. First you work to maintain the distance that you want to fight at. Usually each fighter is working to establish a different range, one that gives them the advantage. Then they are looking for location. Do you want to be in the center of the ring, on the ropes or have the other guy on the ropes or in the corner? All of these positions have advantages and disadvantages. If you are in the corner, you work to advance to the ropes, then to either reverse the position (sweep???) and put the other guy on the ropes or you can escape back to the center of the ring. Once you have your range established, as opposed to the other guys range, you work on trying to set up your technique, getting your angle for your punch or combination to land. Yes, these angles tend to last a lot shorter of a time than a position in grappling... but they are just as important. And there is just as much work done to get to these positions... its just done differently. But the range, and location of the fight, those are things each fighter is always trying to gain and keep, for the duration of the fight... just like taking the other guys back in grappling, if you can get it, you keep it, while the other guy tries to advance his own position out of it.