Hi, frujin, and welcome to PF!
Your question looks simple, but it actually hides a considerable amount of complexity. Rather than trying to answer it directly at once, let me first explain briefly how General Relativity (GR) describes the "gravity pull" of an object.
The central equation of GR is the Einstein Field Equation (EFE), which can be quickly summarized in the popular phrase "spacetime tells matter how to move, matter tells spacetime how to curve". "Spacetime" in this equation is described by something called the "Einstein tensor", and "matter" is described by something called the "stress-energy tensor" (SET). The "mass" of the object is part of what is described by the SET.
The key thing about this equation is that it is the same regardless of what coordinates you express it in. For example, if some massive object is moving relative to you, the EFE looks the same whether you look at it in coordinates in which you are at rest, or coordinates in which the massive object is at rest. (By "looks the same", I don't mean literally that every number in the equation is the same--I just mean that the structure of the equation is the same, and there is a well-defined way to transform the specific terms in it from any set of coordinates to any other set.)
So in this sense, the "gravity pull" of the massive object doesn't change when you change coordinates. The specific components of the SET will change, so it will look like the "mass" changes if you just look at particular components, but the overall prediction for how objects will move in the gravity field of the object doesn't change.
There are also other ways your question could be interpreted, but I'll let you respond to the above first.
