There are several ways to create these "morphing" types of animations. One of the fastest ways to morph complex shapes with different numbers of vertices is to use the blend tool in Illustrator, expand the blend, then release to layers - sequence, then drag the new layers above the original layer. You then import as a comp, trim the layers to 1 frame, then sequence the layers. When I am faced with this kind of a task I usually pick a number of steps in the blend that is about 50% higher than the duration I expect to need for the project to allow for some editing, then nest my morph comp in another comp and use Time Remapping to edit the timing. Here's a very simple tutorial on that technique and it would work very well for animated maps.
Here's another one using more complex shapes:
If your artwork has a similar number of vertices, the same is best, then you can create two or three different shapes in Illustrator on import the AI file, then convert each vector layer to a path in After Effects. When you have the path created you can pick one of the paths, set a path keyframe, then move down the timeline and set a keyframe for the on the second path, then paste to the first, then repeat to complete your animation. The limitation with this technique is that the first vertex must stay in the same position for all paths you create, the number of vertices must be close to the same, and you must have a design that will work when the points move in a straight line. Unfortunately AE's only native tool that can help with complex path transitions is Mask Interpolation and this can only be used on mask paths. You can always do your original work using masks and then convert the animated mask to a shape path if you need a shape layer, but this starts to get tedious very quickly.
Another option is to use 3rd party effects like RE:Flex - RE:Vision Effects to morph images in AE. AE also has some distortion tools that may work for you like Reshape - which uses two masks, and the rest of the distortion tools. Probably my most common distortion tools are Mesh Warp, Liquify and Displacement Map. I also use FreeForm Pro a lot. There's a less expansive FreeForm also available.e. They produces 3D bending effects but can can also be used for 2D work. For morphing lines I often use Trapcode 3D Stroke. The right tool depends on the job.
I'll give you one more suggestion. The TED animations you used as an example could conceivable be created in a single AE comp but the process would get extremely complex and your productivity would slow way down very quickly. AE, in the current form, is a tool primarily designed for creating shots. The process of animation has always been more efficient if each element is a separate little project. I would break your project up into smaller pieces. For example, the morphing map shot at 2:00 in your first example would be a separate comp. I might even have the animating text and the background on a different comp. Most of the shots in that animation cover about one sentence. Some cover a phrase. I would create a separate comp for each of these shots with a few extra frames at the head and tail, render all of the comps, and then edit the project in Premiere Pro. This sounds like more work than doing everything in a single comp or nesting a bunch of comps in your main comp, but it is not. In almost every case AE works better and you are more productive if you do not try and edit a movie in AE.
Thank you very much for such a detailed explanation!)) I really appreciate i!