Can I carry over my character from Edge of the Empire to Age of Rebellion?

As long as your GM agrees, yes. All three games are completely compatible (Han [EotE Scoundrel], Leia [AoR Ambassador] and Luke [FaD Starfighter Ace] on one team, for instance) he just needs to decide how to handle Obligation and Duty (there's sufficient guidance that this shouldn't be a massive problem).

you absolutely can. Each of the three lines (Edge of the Empire, Age of Rebellion, and Forces and Destiny) are meant to work well together. The lines are meant to play easily so that a party can be made up of classes and specialties from any of the books.

That being said, if you want to 'convert' your EotE character to an AoR character, you can do that as well, just finding a class and specialty that works for you, but you shouldn't really need to, you can easily just adapt your EotE character to now be working for the Rebellion, maybe taking a specialization in one of the AoR careers.

Can I carry over my character from Edge of the Empire to Age of Rebellion?

Definitely.

Point of fact, I'm playing what is an "Age of Rebellion" game where not a single one of the active PCs was built using the Age of Rebellion careers, with three of the four initial characters being EotE careers (Colonist, Explorer, and Bounty Hunter) with the fourth being a Commander (mine) who later perished and got replaced with a FaD Warrior.

I've also run some very EotE-themed sessions for my Force and Destiny group, which itself has a PC that was built using EotE (Smuggler/Gunslinger), so as was noted, there's a lot of cross-compatibility between the three product lines.

There's a lot of crossover potential there, really just need to discuss with the GM how to transition from Obligation to Duty, or perhaps combine them, or maybe even have both of them that you would need to balance.

There's a lot of crossover potential there, really just need to discuss with the GM how to transition from Obligation to Duty, or perhaps combine them, or maybe even have both of them that you would need to balance.

One suggestion I've seen is to follow the Han Solo model, and that allow the PCs to completely resolve their Obligations (reducing them down to zero) as part of the early stage of their new lifestyle as Alliance operatives.

Another possible idea is to allow the PCs to use the group's initial contribution rank increase to reduce everyone's Obligations by 10 points. Which if the PCs have been working to reduce their Obligations already, a 10 point drop should be enough to clear most of them off the table.

I'm not an expert on how mechanics will work, but i think the universal specialization "Recruit" serves mainly to this end:

any characther from EotE and FaD can buy the "recruit" and become a good AoR member

of course if the GM allows it, is not really necessary: a bounty hunter or an hired gun from EotE can easily fit inside Alliance army, and a story/roleplay can let you gain membership; however, if you feel career/specs symbolizes "personal life path" (or if the GM is not eager to accept FaD/EotE into AoR campaign), recruit is an elegant way to say "hey i'm a rebel now"

the only substantial difference is Duty/Obligation, but there are rules for them: you can use both or just one. IMHO the best thing is to let characther pick a Duty at starting value (5? even less) and let them fulfill their obligation (and maybe don't get anymore).

Morality, i'd keep them for all Jedi/Force User char, just to track Dark Side/Light Side

Truthfully, any of the EotE careers can work within the structure of the Alliance.

Not every member is going to have a military background after all. A Colonist can be just as vital to the war effort as a Soldier or Commander, though simply in a different way, and are close cousins to the Diplomat. And Technicians are always handy to help keep the mechanical side of things running smoothly for the Rebels and not so smoothly for the Imps. Smugglers have a lot of versatility that can serve the Alliance in a number of different roles, and being familiar with the shady side of the galaxy is useful given how often the Alliance as to deal with and operate in that realm.

All my characters are using Force and Destiny and either AoR specializations or EoE specializations. I have each of them running with morality and they have to choose either an obligation or duty as well.

In my group of Rebels, out of five players two are straight EoE (Explorer/Fringer and Hired Gun/Bodyguard with a secondary specialisation of Gunslinger), two are straight Aor (Engineer/Saboteur and Soldier/Sharpshooter with a secondary specialisation of Medic) and one mixed (Ace/Pilot from Aor with a secondary specialisation of Force Sensitive Exile from EoE). No compatibility issues at all. Everyone has a Duty as they're members of the Alliance, and any who wanted extra XP or cash at character gen could use either the reduced Duty rules from AoR or the increased Obligation rules from EoE.

The Edge of Darkness campaign that is on the boards has my character built from EotE (Geoff Hanson, Colonist: Doctor). The idea here is that he is part of the Refugee Relief Movement (briefly mentioned in Far Horizons). The RRM is a combination of the Red Cross and Doctors without Borders. Geoff Hanson was part of it until Alderaan was destroyed, as well as other issues. The reasoning for him to join was that he was tired of the Empire causing more harm than good, especially with their heavy handed tactics and interception of supplies meant for the civilians.

Over all, it seems to be enjoyable by the party and GM, and I enjoy playing him, as well.