First King of Ganga Dynasty started in Karnataka when Jain Acharya Simhanandi inspired his two disciples Daddigh and Madhav to establish their rule, which they did by constituting the territory of Gangawadi with Kolar as their capital. Madhav Kongunivarma I was the first crowned king of this dynasty, who ruled for a long period during 189-250 AD. So the question of Gangas originating in north and coming to south is irrelavant. They actually moved south(3rd to 10th century ) to North(11th to 15th century) to establish Eastern Ganga Dynasty. The Eastern Ganga dynasty established with decline of western ganga dynasty. They built temples in orissa on similar lines as in Karnataka. In the opinion of Dr. N. K. Sahu, both the western and eastern Ganga dynasty belong to one and the same dynasty.
Mr. Dubey has identified the Tumbura-race with Mahisa, Thumbra being mentioned in Puranas as Ganga. Mahisa race as we know is from only one place , that is mysore from ashokan times . The first king of this Ganga dynasty Anantavarma belonged to the Mahisa race as per Puranas. Kudlur grant of Marasimha and the Santara inscription on the Huncha stone says Simhanandi gave them kingdom. The last mentioned record indeed refers to him as "the archariya who made the Ganga kingdom." :"Ganga-rajyaman madida Simhanandy acharyya." which shows they are of kannada decent or Mahisa race. The Western Ganga rule was a period of brisk literary activity in Kannada, which shows that they are kannada origin.
Eastern Ganga capital is kalinganagara, which is often leading to misunderstanding it is in kalinga(orissa), but it is actually in Andhra pradesh. The Eastern Ganga dynasty came to prominence after 10th century. They maintained relations with cholas( part of chalukyas) shows that they are of same as western Gangas. The Karnata race having independent rulers in various places of India in the previous thousand years has established a kingdom in orissa and Andhra. With Chalukya's in West and South, senas in Bengal , srilanka, Nepal. The karnatas ruled whole of India. Showing eastern Gangas belong to fisherman decent of orissa is not correct. Gangas continued the tradition of building temples in orissa also with Sun temple of Konark , Lord Jagannath temple and various other temples. Many of the common Orissan surnames, such as ‘Dalai’ and ‘Senapati’, originated in Ganga times. Gangas also took the mahisasura mardini or Durga worship to orissa from karnataka.
Origin of Gangas is mainly due to confusion from Puranas. Whose dating has always been in question. Puranas show 80kings ruling 1600years. Also the opposition of puranas to Jain territories mainly in karnataka means , karnataka decent does not get due respect. There is also mention of Ganga Vamsa apart from Ganga dynasty and also Mahisa race.
Historical sources are not very clear from sanskrit inscriptions and plates give no continous evidence of any known decent except they are Ganga rulers. Historical records are in sanskrit with mix of southern and northern script , which shows that the western and eastern ganga's are same.
External sources like arab show that the eastern ganga land was essentially agararian and army infantry. So the question of gangas moving from karnataka to orissa is not a issue. Also the southeast asian sources like indonesia tell about karnataka traders operating from eastern shores.
So the eastern and western Ganga are all same. Ganga origin is in Mysore( Karnataka). when the primacy of Ganga rule came to end in karnataka came to end around 10th century, they moved north to andhra and Orissa to establish rule. They constructed many famous temples and also increased trade there.