It's pretty simple.
The formal definition of velocity is the derivative of position with respect to time. So in one dimension:
$v = \frac{dx}{dt}$
And therefore
$x(t) = \int_{0}^{t}v(t')dt'$
If you assume $v(t)$ is constant, then that equation becomes $x = vt$ (hence the kinematic equation). If $v(t)$ isn't constant, then $x=vt$ is almost certainly not correct.
