[HopDavid]

Altitude of EML1 is about 320,000 kilometers. An ordinary 320,000 x 300 km earth orbit would be moving 10.81 km/s at perigee.

As an object falls towards earth it will surge ahead of the moon and the moon will be pulling it backwards. This makes effective apogee more in the range of 300,000 km.

A 300,000 x 300 km orbit has velocity of about 10.8 km/s at perigee.

In other words, dropping from the neighborhood of L1 will still give you speed very close to escape velocity by the time it reaches earth.

In the case of orbital tether, an elevator car would have gripping wheels in contact with the tether. If its's descending the wheels can power a generator to charge the elevator car's battery.