If we consider a point on the central part of a contracting object as observed from earth, the point is continuously moving away from us as the object contracts. Moreover, light emerges from a deeper gravitational well as the object contracts. So, shouldn't contracting objects show a redshift? And if so it should be less towards the edges than towards the centre, right?
Contracting objects can and do exhibit a measurable red shift, and this has actually been measured.
For example, Cepheid variable are stars that expand on contract in a cycle lasting days to months.
This paper by R. Paul Butler discusses the measurement of Doppler shifts in Cepheid variables. According to the paper, radial velocities as small as 3 meters/second have been measured. Cepheids have been measured to have Doppler velocities of 30 to 1000 meters/second.
The paper was published in 1997. It says that the precision of such measurements had improved by more than two orders of magnitude (a factor of 100) in the preceding decade. I presume the technology has continued to improve in the nearly two decades since then.
shouldn't contracting objects show a redshift?
Yes, they should. But do they, actually? That's the better question.
To have any appreciable redshift, the speed of the object needs to be huge. That will be a very short implosion. Also, there aren't many mechanisms that can accelerate implosions that much.
What I'm saying is - it's theoretically possible, but in practice you won't see it that often, if ever.
And if so it should be less towards the edges than towards the centre, right?
Sure. If the object is transparent to its own radiation, then you'd see blueshift from the other side, too. Again, this is very theoretical.