Liquid-liquid caffeine extraction
I suggest to use a 1:1 mixture of dichloromethane and 0.2 M NaOH at room temperature.

The protocol was described by Tetsuo Onami and Hitoshi Kanazawa in J. Chem. Educ, 1996, 73, 556-557.

The authors mention that only trace amounts of caffeine are extracted from tea leaves with dichloromethane alone. They reason that due to the hight amount of tannines in the tea leaves, most of the caffeine exist in protonated form and is therefore bound to the tannines.

As an experimentally proven alternative, they suggest to treat the tea leaves with a mixture of dichloromethane and 0.2 M NaOH at room temperature!

According to the article, 20-30 mg of caffeine were extracted from a 2 g tea bag after shaking the above mixture for about ten minutes.


UPDATE
If you insist on the liquid-liquid extraction of previously prepared aqueous solutions and feel the urge to set up a percolator-type apparatus for continuous extraction, the Kutscher-Steudel glassware will not work with solvents that have a higher density than water.

For continuous extractions with solvents like $\ce{CH2Cl2}$, some different glassware has been designed and was described by S. Wehrli in Ein Apparat zur Extraktion von Lösungen mit schweren Lösungsmitteln and was published in Helv. Chim. Acta, 1937, 20, 927–931 (DOI)