This my explanation on how Sasuke's and Kaguya's Sharingan abilities work and how they are transporting through dimensions and such.
Chapter 685 showed us some of Obito's abilities using both of his Sharingans, however we also got a direct comment about Sasuke's abilities from himself. He comment that he was able to switch places with another object in an instant, but it was limited for its short reach. However, in the Mangapanda translations, it was stated that Sasuke is able to travel through dimensions far from him using his eye, but that goes against what he said about the limited reach and it was not even mentioned in the Mangastream translation. Though one thing both translations had in common was that Sasuke was able to sense Naruto's chakra perfectly in a specific place, this is as much as a lead to Kaguya's abilities as anything we've gotten so far. Let me explain the actual theory here.
There is a common ideal in our reality that there are more dimensions out there that are alike our own, to the point that they are completely alike with only a single new variable. I think this concept is how Kaguya's and Sasuke'sSharingan jutsus works. Sasuke explained that he is able to swap places with something in an instant, but I don't think he is only able to swap places with himself and another object. I think he is able to swap places with anything within his reach. I also think that this is why each of Kaguya's dimensions have a castle within them, and I will get back to that later.
When Sasuke first performed his new jutsu to counter Madara's attack, he most likely swapped places with himself and a little rock. When he stabbed Madara with his sword using it, he most likely swapped places with himself and his blade. When he and Naruto tried to seal Madara away, he didn't swap places for him and Naruto, he swapped Madara this time. This was later repeated when they tried to do the same with Kaguya, and it was shown better with her. When they tried to use it on her, the lines symbolizing attention was show to the center, meaning that neither of the two were moving. However, Kaguya was flying away from Naruto's punch and when they were about to do it, she had slight horizontal lines covering her, this symbolizes movement in a manga. Meaning that she stopped moved very quickly, most likely because Sasuke teleported her.
This fits within the theme of Uchiha's manipulation imagination itself in their jutsus. It also fits with Sasuke's character as he is very talented in Nature Manipulation and Shape Manipulation. Sasuke, right now holds the power to manipulate the battlefield to his liking. Meaning that if you don't have a higher reaction speed than Sasuke, then you won't be able to escape him if you are within his reach and you will have a hard time fighting him in general. Its also likely that Sasuke is able to transport jutsus to his liking, so missing a ninjutsu for him isn't such a big fail now. However, I think that this is only a weaker version of the Kaguya's ability.
I believe that Kaguya's ability is yet another "swap places" jutsu, same as Sasuke's one. But it is certainly a bit different. In all of Kaguya's dimensions which she's been transporting the others two using her instant teleportation, its been a castle. The Desert and Acid world most likely has one too, but I don't think her main world has one. (The one she and Obito's teleportation got them to.) I think that these castles works as a sort of "anchor" or "connection" between the worlds. She is not teleporting between dimensions when she does it instantly, however she is teleporting through them when she uses the black gates to do it. (Like in herself and Sasuke's case.)
I believe that the Sharingan carries three main abilities to it. The regular 3 Tomoe Sharingan has the ability to see things as if they were going in slow motion, because of this they are able to see movement such as Hand Seals and acrobatic movements in great detail and are able to copy them. The 6 Tomoe Sharingan holds the power to swap places between two things. (This is the ability Sasuke and Kaguya has been using.) The 6 Tomoe Sharingan holds a yet unknown power that most likely has something to do with Kaguya's plan, do it may be that this only holds the power of Mugen Tsukuyomi. Now, lets have a little story telling about how I think Kaguya achieved these powers and how they fully work.
Time/Dimensional Travel & Dimension Swapping
When Kaguya obtained the power of the Shinju, she gained the power to open gates to any dimension she wanted. However, in order to travel to a dimension she needs to travel to the main one which is connected to the other ones. (The one that Obito, Sakura and Sasuke currently are located in.) Dragging people through this black gates an into dimensions aren't a very useful tactic in combat as it uses far too much chakra and its possible to dodge. However, I think Kaguya improved this ability through the power of the 6 Tomoe Sharingan as she has the 9 Tomoe Sharingan.
When Kaguya gained the ability to open the black gates that connects the worlds, she entered one dimension after the other in order to build a castle in each of them. Upon doing so she preformed the power of the 6 Tomoe Sharingan to swap places between the dimensions at will, dragging anyone she wishes into it. We know that Sasuke has a limit to how far away he can swap something, I do believe that Kaguya also holds this weakness and I do believe that she is no able to decide who not to swap into her dimensions. In other words, every person within her reach will be swapped into her dimension at choice. The reasons I state this is that she had no reason to transport anyone else besides Sasuke and Naruto, yet she did. She also used the Black Gate to transport Sasuke alone, instead of using the instant one, because she most likely aren't able to swap with only 1 person through dimensions.
But how does "swapping dimensions" make any sense at all? Well its simple, she has the castles in each dimensions to act as a "anchor" or "connection". I also believe that each dimension she has exists in the exact same location as the others, just that they are in a different existence. The reason I say this is that Sasuke was able to sense Naruto's chakra in a specific area, as if it acted as a gateway to the other dimension or as if in another existence, Naruto were in that location. Just like the Kamui dimension, the exact coordinates in the real world are the same location in the Kamui dimension. The same goes for Kaguya's, the coordinates in one world is the same as in another only that the worlds have different landscapes.
Because of this, she is not teleporting through dimensions when she changes them, she is swapping the world inside her reach into another dimensions using the castles as "connections" between them instead of a object. This means that destroying each castle will make it impossible for Kaguya to instantly swap dimensions, meaning she would have to use the Black Gates to do it manually.
Why don't you add pictures? >.> I don't believe Sasuke's and Kaguya's Rinnegan abilities are connected. I don't think you can put them on the same level.
Because there is little point in adding pictures, and the main reason is that people who really are interested would go and check out the pages themselves. (Meaning that those people would actually think about their counter-arguments and actually care about them, unlike others.) Also, I do think they are using the exact same jutsu, only that Kaguya uses it on a much larger scale thanks to all the chakra she has.