Read my post on the technology used in the Pushpak Vimana.
What route did Ravana take while kidnapping Sita and how did he move her in Lanka?
As far as I could check information as to how Sita was abducted is limited Ravana taking her in Pushpak Vimana and later fighting with Jatayu and reaches Lanka, in Valmiki Ramayana.( Corrections to this welcome with slokas).
But Buddhist Legends of Lanka give a detailed description about how Sita was moved in Lanka.
“this vehicle landed at Werangatota, about 10 km from Mahiyangana, east of the hill station of Nuwara Eliya, in central Sri Lanka. Sita was then taken to Goorulupota, now known as Sitakotuwa, where Ravana’s wife, Mandodari, lived. Seetakotuwa is about 10 km from Mahiyangana on the road to Kandy. Sita was housed in a cave at Sita Eliya, on the Colombo-Nuwara Eliya road. There is a temple for her there. She is believed to have bathed in the mountain stream flowing beside the temple.
North of Nuwara Eliya, in Matale district, is Yudhaganapitiya, where the Rama-Ravana battle took place. According to a Sinhalese legend, Dunuwila is the place from where Rama shot the ‘Bramshira’ arrow that killed Ravana. The Sri Lankan king was chalking out his battle plans in a place called Lakgala when the killer arrow struck him. Lakgala is a rock from the top of which Ravana could see north Sri Lanka clearly. It served as a watchtower following the expectation that Rama would invade the island to rescue his consort. Ravana’s body was placed on the rock at Yahangala for his subjects to pay their last respects. Since Ravana was a Brahmin, it was considered a sin to kill him, even in battle. To wash off the sin, Rama performed puja at the Munneswaram temple in Chilaw, 80 km north of Colombo. At Manaweri, north of Chilaw, there is a temple gifted by Rama.
According to another legend from the southern part of Sri Lanka, Sita was actually detained in the mountainous forest area of Rumassala near Galle. When she fell ill, Hanuman wanted to bring some medicinal plants from the Dronagiri mountain in the Himalayan chain to cure her. Since he could not find the plants, he brought the whole mountain and dropped it at Unawatuna, which is near the present Galle harbour. Unawatuna means ‘here it fell’. Indeed, the area is known for its medicinal plants.
At Ramboda, in the central highlands, known for its massive waterfalls, a temple for Hanuman has now sprung up as the belief is that he had visited Sita who was incarcerated there. Legend has it that the Koneswaram temple, in the eastern district of Trincomalee, was gifted by Lord Shiva to Ravana, as he was an ardent devotee. In the famous Buddha Vihara at Kelani, near Colombo, there is a representation of Rama handing over captured Sri Lanka to Ravana’s brother, Vibheeshana, who sided with him in his conflict with Ravana._Dandu Monara Yanthraya.
Ravana using a golden deer as a decoy visited Sita when she was alone in the guise of an old sage, abducted and brought her to Weragantota in Lanka in his plane, the Pushpaka Vimana.
Weragantota means the Place of Aircraft landing in Sinhala. This is the first place Sita Devi was brought to Lankapura (capita city of king Rāvana).
Sita Devi was kept at queen Mandothari’s palace at Lankapura. The place Sita was held captive is called Sita Kotuwa (also known as Seetha Kotuwa); which means Sita’s Fort in Sinhala. It is believed Rāvana had an aircraft repair centre atGurulupotha close to Sita Kotuwa. Gurulupotha means Parts of Birds in Sinhala.
Ravana moved Sita from Sita Kotuwa to Ashok Vatika the salubrious garden in the mountains. The route too was said to be spectacularly beautiful, as Ravana wanted to show Sita the beauty of his kingdom.
The Chariot Path atop the mountain range is still visible. The Sita Tear Pond close to the Chariot Path is believed to have been formed by Sita Devi’s tears. Visitors could also see the famed Sita Flowers which are endemic to this area.