Suppose a man is running and he gradually speeds himself up . For this he applies a force on the ground backward and the ground pushes him forward . This is probably due to to friction between the shoes of the man and the ground . The friction acting in this case is Kinetic friction which has a constant magnitude. Since the magnitude of frictional force is constant, how is the man able to accelerate himself ? Who is providing this force?
Don't think kinetic friction, just because some part of him is moving. Is it also kinetic friction if I stand still but swing my arms? Many particles in him or elsewhere might or might not move. They are irrelevant.
Only the particles in contact with the ground are relevant.
And they are not moving. His foot is not moving during the step. It is stationary and not sliding while touching. There is static friction here.
And static friction can vary easily.
Is the shoe slipping? If not, then it is not kinetic friction. If the shoe is not slipping, the bottom of the shoe is stationary with respect to the ground and the force being applied is static friction. Unless he's running on ice or a lubricated surface, it's unlikely he's experiencing kinetic friction.
The force of friction is not constant in this case. The force of static friction is only what is necessary to prevent slipping. Over the course of his stride, the force of friction changes.
Acceleration occurs when the net force on an object is non-zero. Nothing else matters. It doesn't matter if the force is constant. It doesn't matter how many forces act on the object. It doesn't matter if the object is moving the same direction as the force. As long as the net force is non-zero, it will accelerate. In this case, there are only three objects acting on the runner: gravity, friction, and the normal force. Gravity is directed straight down, and the normal force is directed straight up (assuming flat, level ground). Friction acts horizontally to accelerate the runner.
For part of the stride the man is certainly exerting a force on the ground in the backward direction and the ground is exerting a force on the man in the forward direction which causes the man to increase his momentum and hence his velocity in the forward direction.
The frictional forces exist because the muscles of the man push the foot into the ground.
It matters not whether his foot is slipping or not relative to the ground except that a greater frictional force can be exerted on the man if there is no slipping.
So you have a series of impulses (force $\times \Delta$time) acting on the man due to the ground which increases his momentum and velocity.
I think that the interesting fact is that for the other part of the stride the frictional force acting on the man due to the ground is actually in the opposite direction to that of his motion and imparts an impulse which tends reduce his momentum.
That is why a man can have a positive acceleration, move at "constant" velocity or have a negative acceleration.
It all depends on the magnitude and direction of the frictional force which on the mas at various phases of his stride.
You might find this article informative from which I obtained this graph which shows the force exerted on a person (including in the vertical component) when jogging.
In particular note the dashed graph which shows the horizontal (frictional) forces acting on a person who is jogging which changes direction during a stride.