The tesseract was a piece of artificial technology built by the beings in the higher dimension (the 'bulk', as physicists term such a possible higher spatial dimension), and part of its design was to assist Cooper in communicating with people in the past. For example, when he banged on the books he saw in Murph's childhood room, it wasn't the natural gravity of his own body that caused the books to move, rather the tesseract was artificially creating gravitational waves that traveled back in time to move the corresponding book. In one example in ch. 30 of The Science of Interstellar by physicist Kip Thorne, we see that the tesseract continues to generate a repeated message on Murph's watch via gravitational waves affecting its hands, even though Cooper only actually sent the message once, making clear that this isn't just a matter of his body's gravity field directly affecting the watch:
By the time Cooper has received the quantum data from TARS, he has
mastered this means of communication. In the movie we see him pushing
with his finger on the world tube of a watch's second hand. His pushes
produce a backwards-in-time gravitational force, which makes the
second hand twitch in a Morse-encoded pattern that carries the quantum
data. The tesseract stores the twitching pattern in the bulk so it
repeats over and over again.
Since the tesseract is artificially helping him communicate, and is either taking him to people the higher-dimensional beings want him to communicate with or reading his thoughts about who he wants to see, it shouldn't be any problem for it to match speeds with the ship that Amelia Brand is riding in (as Liesmith said, there is no absolute notion of 'rest' in relativity, so this would require no more or less finessing than keeping the tesseract at rest relative to Murph's bedroom), or to artificially create a localized gravitational distortion that bends light in a way that she can see and exchange a "handshake" with.