After Effects is changing the Frame Rate on Imports from Premiere Pro.... should I change back?

FRAME RATES FOR COMPS AND PPRO SEQUENCES

The frame rate for your sequence as shown above is 23.976. This is one of the three standard frame rates in all NTSC countries that have 60Hz power. You should always use these frame rates if your intended audience is ever going to see your work on any television. I will explain why later. There are only a few theaters left in the world that have film projectors. The standard frame rate for film projection is 24 fps. In the highly unlikely chance that you are transferring your video production to film the standard frame rate for your sequence (Premiere) or your comp (After Effects) would be 24 fps. This is the ALMOST the ONLY time it makes sense to set your frame rate to 24 fps.

Europe and a few other parts of the world operate on PAL 50Hz power. The standard frame rate for all broadcast video and even film projection is 25fps. That should be the frame rate of almost all of your standard video production.

FRAME RATES FOR FOOTAGE

The frame rate for your footage is 29.97 fps. This fractional frame rate standard was established when Color Television was introduced. Before color TV the frame rate for all broadcast tv was 30 frames per second interlaced. Almost no one uses interlaced footage any more because all of the equipment we use to create and playback video is fast enough to draw every row of pixels in one cycle (60Hz). When color TV was developed in NTSC land, they had to slow down the frame rate just a little so they could include the color information in that .03 part of the frame. 

The frame rate used in the camera has nothing at all to do with the playback speed. If you playback a video or movie at the same frame rate used to shoot the movie, playback will be in real-time. If you want to change the speed of the movie you need to run the video at a different speed. You do not do that by changing the frame rate of the comp or sequence, you do that by changing the frame rate of the source video. 

Most video cameras shoot at 29.97 fps in NTSC land or 25 fps in PAL land. Almost all video cameras are capable of shooting at two or three other standard frame rates. 23.97 and 59.94 for NTSC and 50 for PAL. Mobile devices usually shoot at variable frame rates somewhere close to those video standards. Professional cameras are capable of shooting at almost any frame rate from 1 or 2 fps up to 120fps or more. High-Speed cameras are capable of up to 20,000 fps. Once again, unless you specifically tell your system to playback footage at a different frame rate than was used to shoot the footage, the playback will be in real-time.

Different frame rates give a different look to the final video. The higher the frame rate the lower the motion blur. Put your hand out in front of your face as far away as you can, focus on something on the other side of the room and move your hand left and right. The motion blur your brain applies to your hand is about the same as the motion blur you get at 24 fps when using a 1/48 second exposure time or a 180º shutter. That is why 24 to 25 fps was picked as the standard frame rate for motion picture production. This gave a smooth appearance to the action and, because of the size of the standard 35MM film frame, there was also enough room on a frame of film to provide decent audio.

INTERPRETING FOOTAGE

Both After Effects (your third screenshot) and Premiere Pro (your second screenshot) are properly interpreting the footage. The original footage was shot at 29.97 fps. That means the playback speed will be accurate. There are no errors here. Unless you want to force your original footage to playback faster or slower than real-time, do not mess with these settings.

DYNAMIC LINK

When Premiere Pro creates a new After Effects comp from selected footage that comp is created at the frame rate used by the sequence. It is the only thing that makes sense to do. You want to replace every frame in the sequence with the new frames from AE. The footage that is imported into AE is also brought in without changing the frame rate of the footage. It has to be. If it was not, the playback speed would change.

When you put your 29.97 fps footage in a 23.976 fps sequence in Premiere Pro you end up with some blended frames. It is hard to see the blended frames when the video is playing back, but they are there. The same things happen in After Effects. Your comp has 29.97 fps footage in a 23.976 comp so the frames are blended. 

When you go back to Premiere Pro and look at the linked comp, the frame rate of the comp is also correctly reported as 23.976. You did something to the frames in After Effects so now every single frame you modified in AE is in Premiere Pro and there are no Blended new frames. 

There will be no timing problems with your workflow, and if you are happy with the blended frames in Premiere Pro when you don't match the sequence frame rate to the source footage frame rate then nothing is lost.

If you are doing visual effects and you want absolutely the highest quality from every frame you should be matching the Premiere Pro sequence frame rate with the footage frame rate. That is the more professional way to work. I always do my VFX work in comps that match the frame rate of the footage because I want every original pixel available. I almost always use 29.97 fps for my projects unless they are destined for theatrical distribution. 

I hope this helps. If it is any consolation, about 80% of the people I work with that have never shot with a film camera do not correctly understand frame rates, motion blur, critical panning speeds, or how to shoot footage so it is easier to do visual effects. Everybody I know that learned their craft using film cameras understand because there was no metadata telling us what frame rate was used to shoot the footage.