Ciao Michele
Not sure about first four weeks of CRUSADER, but from mid-December to mid-January Lorraine was quite active in hammering the encircled Italo-German troops at Bardia and Sollum/Halfaya. They also took losses during this period, to AA, IIRC.
There is a book in French "Croix de Sud, Croix de Lorraine", about the free-French air force in North Africa 1941/42, which may interest you, if you speak French.
http://www.france-libre.net/fafl/uni...e-lorraine.php
Cordiali saluti
Andreas