In an affinely connected spacetime with a metric compatible connection, the equation of the curve in which the tangent vector at each point is the result of the parallel transport of every tangent vector to the curve along the curve (to that point) is:
$\dfrac{d^2x^\lambda}{d\tau^2} + \dfrac{dx^\mu}{d\tau}\dfrac{dx^\nu}{d\tau}\Gamma_{{\mu}{\nu}}^{{ }{ }{\lambda}}=0$
Where $x^\lambda$ represents the coordinates, $\tau$ is the affine parameter, and $\Gamma$ is the affine connection. We call such a curve an auto-parallel curve.
Now, the equation of the curve which extremizes the action (where action is defined as $S=\int\sqrt{g_{{\mu}{\nu}}dx^{\mu}dx^{\nu}}$) is given by:
$\dfrac{d^2x^\lambda}{d\tau^2} + \dfrac{dx^\mu}{d\tau}\dfrac{dx^\nu}{d\tau}\begin{Bmatrix} \lambda \\ \mu\nu \end{Bmatrix}=0$
Where $g_{{\mu}{\nu}}$ is the metric (and the connection $\Gamma$ is compatible with it) and $\begin{Bmatrix} \lambda \\ \mu\nu \end{Bmatrix}$ represents the Christoffel symbols. We call such a curve a geodesic curve.
If we impose the condition that the torsion must vanish then the two of the above equations represent the same class of curves but otherwise, they represent different classes of curves. So, in a generic case where torsion might be non-vanishing, should the trajectory of a free particle be described by the auto-parallel curves (which I feel should be the answer according to the Principle of Equivalence) or by the geodesic curves (which I feel should be the answer if we respect the Principle of Extremum Action)?