Great question and this is a famous confusion in special relativity. In our frame of reference, it becomes more and more difficult to accelerate a particle, and infinitely so as it approaches the speed of light. It seems as if the particle is gaining inertia -- its mass trending to infinity. But it isn't. This is simply a property of the Lorentz transformation between the two frames of reference. It is the same particle, with the same mass, and according to its clock and meterstick it is accelerating just fine and not growing infinitely overweight thankyouverymuch.
The confusion arises because [math] is the internal energy, where [math] is the rest mass -- a property of the object itself and independent of its speed. You can increase the internal energy of something by, say, heating it up (raising the thermal energy), which will increase its rest mass. In other words your mass measures your internal energy! But the internal energy does not change when you speed it up. A simple reason is that motion is relative. There is no local experiment that you can do that will tell you how fast you are moving -- your motion only makes sense with respect to something else. Therefore your speed is not an internal property, and your kinetic energy cannot affect your rest mass.
The total energy (internal plus kinetic) in relativity is [math], where γ is the relativistic factor that approaches infinity as the speed approaches the speed of light. (It's 1/sqrt[1-v^2/c^2]). But m is still constant!
It may be tempting to define a "relativistic mass", as [math], to capture this idea that the particle appears to gain inertial mass as its speed increases. But it is a poor choice, and Einstein himself advised against it. There is no need to consider any mass besides the rest mass. All of the properties of the object at varying speed follow directly from there. Then the familiar formulas for energy and momentum become [math], and [math].
So, a black hole moving at near the speed of light still has the same mass and gravitating effect as if it was at rest. If this was not true, then the principle of relativity (that motion is relative and cannot be measured with respect to absolute space) would be violated!