I. Conservation of energy and momentum
If the system is isolated, energy will be conserved.
If no external force acts on the system, momentum will be conserved.
These are the laws of conservation of energy and conservation of momentum, respectively.
You can find proofs for these laws within the context of Newtonian physics in any textbook on classical mechanics.
Notice the importance of deciding what is the system of interest.
For the same physical process, in one system momentum (for example) will be conserved and in another it won't.
Let's examine one example of oscillatory motion in detail.
Suppose we have a mass attached to an ideal spring attached to an infinitely massive wall oscillating horizontally on a frictionless plane.
The system is the mass and the spring.
This system is isolated---the wall is infinitely massive, so its motion isn't affected by spring force.
The spring is ideal so it doesn't even warm up the wall.
In fact, it is straightforward to solve for the motion of the mass and then to calculate the energy explicitly and see that it is a constant.
The momentum of the system will clearly not be conserved since the velocity of the mass is a function of time.
In fact, there is an external force acting on the system, so we shouldn't expect it to be conserved.
Now assume the spring to be connected to some (possible very large) mass that also rests on the frictionless plane.
Let's consider the system of interest to include both masses and the spring.
If we assume this new system is isolated, energy will again be conserved.
As before, one can calculate the energy explicitly to see that this is true.
If we consider the total momentum of both masses taken together, momentum will now be conserved as long as no external force acts on the system.
II. Motion near the equilibrium position
Motion is described vectorially, so it is natural to think in terms of momentum here.
For small displacements near equilibrium momentum will be constant since the spring force at equilibrium is vanishingly small.
This is just Newton's first law.
Thus, the mass just passes by the equilibrium position with its momentum essentially unaffected.
If you prefer to think in terms of energy, the kinetic energy of the mass as it passes through the equilibrium position will also be constant since the change in the potential is vanishingly small.
Thus, the object's speed must remain essentially constant as it passes through equilibrium.