Having just read the attached article from earlier this year in the French press about the Jonathan Coulom case, one part of it piqued my interest and may hold the answers to a few questions about how an abductor could have left through the open window with Madeleine & carry her in the manner Tannerman did.
https://www.lefigaro.fr/faits-divers/affaire-jonathan-coulom-un-pedocriminel-allemand-denonce-par-son-codetenu-20210218.
*snipped*
"The French investigators already had the German lead in mind, because there were very important similarities between what had happened for Jonathan and the little boys killed in Germany by Martin Ney: they were all found with fists and knees tied, in the same fetal position..."
Hypothetically & imo...
It is possible that Madeleine was bound at the wrists and just above the knees then the former bound to the latter to provide almost total restraint. She would not have been able to remove any gag used to silence her. An abductor could have easily passed her through the open window and lowered her to the ground in her trussed state before quickly exiting himself.
Looking at the image of Tannerman, Jane saw the child's knees close together. It would be very difficult to carry a child restrained in the manner described above, both upright & against an adult's shoulder - a more favoured position for carrying a child any distance, especially a sleeping one - as opposed to the carrying style witnessed by Jane.
Equipment required by abductor:- rope, cloth for gag & possibly a utility knife.