Since I work with adhesives like this for my dayjob..I'll shed some light.
Any tape that you can feel the grab, and is "tacky"..is a pressure sensitive adhesive. The definition of a pressure sensitive adhesive is one that will still flow* at room temperature...once its cured or set. Elmers glue, would be an example of something that although an adhesive, is not a pressure sensitive, once cured/dried.
*flow means ability to move....not like flow like a liquid flowing.
The part of the adhesive that makes its tacky, are the side chains off the main backbone of the molecule. Those are what give the adhesive its "tacky" feel.
There are two aspects to any pressure sensitive adhesive.....tack and peel strength. 95 % of the time..these two are inversely proportional. If you have a high tack adhesive, its typically not that strong. Conversely, you can have an adhesive that does not feel that tacky, yet be really strong and have high peel values. There are exceptions to this generalization of course.
The issue with selling to the general public, is they do not understand the above mentioned relationship..and always associate tack with strength. So, suppliers are forced to supply a tacky adhesive, that in reality, may not work as well, as one for an application, where a less tacky one would.
Now, going back to flow...the adhesive used in the pics above..where its sliding..my educated guess would be its seeing heat...above room temp....which of course lets it "flow" more.
The reason this happens more as it gets older, is the nature of adhesive itself changes over time..which typically goes one of two ways..its get "gelly", like in the pics above, or it gets "dryer" and loses its tack over time, but usually gets stronger...think factory stickers..trying to get them off years later.
