Context :

Aims : We derive the vertical velocities of disk stars in the range of Galactocentric radii of R = 5 - 16 kpc within 2 kpc in height from the Galactic plane .
This kinematic information is connected to dynamical aspects in the formation and evolution of the Milky Way , such as the passage of satellites and vertical resonance and determines whether the warp is a long-lived or a transient feature .

Methods : We used the PPMXL survey .
To improve the accuracy of the proper motions , the systematic shifts from zero were calculated by using the average proper motions of quasars in this PPMXL survey , and we applied the corresponding correction to the proper motions of the whole survey , which reduces the systematic error .
From the color-magnitude diagram K versus ( J - K ) we selected the standard candles corresponding to red clump giants and used the information of their proper motions to build a map of the vertical motions of our Galaxy .
We derived the kinematics of the warp both analytically and through a particle simulation to fit these data .
Complementarily , we also carried out the same analysis with red clump giants spectroscopically selected with APOGEE data , and we predict the improvements in accuracy that will be reached with future Gaia data .

Results : A simple model of warp with the height of the disk z _ { w } ( R, \phi ) = \gamma ( R - R _ { \odot } ) \sin ( \phi - \phi _ { w } ) fits the vertical motions if \dot { \gamma } / \gamma = -34 \pm 17 { Gyr } ^ { -1 } ; the contribution to \dot { \gamma } comes from the southern warp and is negligible in the north .
If we assume this 2 \sigma detection to be real , the period of this oscillation is shorter than 0.43 Gyr at 68.3 % C.L .
and shorter than 4.64 Gyr at 95.4 % C.L. , which excludes long-lived features .
Our particle simulation also indicates a probable abrupt decrease of the warp amplitude in a time of about one hundred Myr .

Conclusions : The vertical motion in the warp apparently indicates that the main S-shaped structure of the warp is a long-lived feature , whereas the perturbation that produces an irregularity in the southern part is most likely a transient phenomenon .
With the use of the Gaia end-of-mission products together with spectroscopically classified red clump giants , the precision in vertical motions can be increased by an order of magnitude at least .