The Lucavi are demons from Hell who are trying to take over the mortal realm. Each one seems to take on some aspect of the stone they are associated with. How they came to be linked to the stones is unnclear; however, I have a very good idea of what happened.
The Zodiac Braves story - the original ones, not the group Ajora OR Vormav had - was most likely true, a group of warriors who possessed these magical holy stones amd used them to seal the twlve demons away - one for each stone. They were kept safe, able to contain the evil power within and keep them from this world. However, the stones were to fall into the hands of others, and most notably Ajora those twleve centuries before our story.
The Lucavi and Ajora are tied if only by associations of the stones, but how much did Ajora know about the Lucavi? It is unknown if the people knew the full truth of the stones - Ramza certainly did not until he saw Queklain, but then at the end of Chapter 3 we see the same stone - Scorpio - revive Malak. Is it possible that the Holy Stones which sealed the demons also contained a power of Good, the power which was used to trap the demons and could be used in such fashion? I'm inclined to believe so, or perhaps on that event it was the death of Queklain - taking away the evil powers/corruption of the stone - which released the good powers within. The stones were considered holy long before the time of Funeral's attempt to use the story for his own purposes; it is not far-fetched to believe that they contained good power as well as the trapped power of the Lucavi.
As for how the Lucavi were breaking free, its possible that after so many years the stones' power was waning, or the Lucavi's desire was becoming strong enough to break through - or perhaps it was Ajora himself before his death. None of this is clear; the past events with the stones is very vague at best. However, this fills the gap nicely, wether Ajora or time did it. Vormav was the first to fall to the Lucavi in the times after the Fifty Year War, as Funeral strove to use the story fo the Braves to win the hearts of the poeple. Vormav - now possessed by Hasmmalum - began to work to ressurect the others, finding suitable "hosts" and to begin the resurrection of St. Ajora and Altima the Bloody Angel.
Ajora's ties to the Lucavi aren't clear, but its likely that he discovered Altima when trying to revive the Braves Story himself - and linked with the Bloody Angel in hopes of eternal life - or the chance to resurrect again - and also for power. Ajora was a man getting into danger and making enemies with powerful people; perhaps it was a bargain with the devil, but he would cheat those who sought his end and get his revenge later. Why would Altima choose someone such as Ajora to link with?
Ajora had a growing base of followers, all people who would one day become followers of the faith he preached - all people who could be potential candiates now or in the future to host the Lucavi. Through means unknown they could draw those they needed, and make the Glabados faith a tool for their own use while its leaders unintentionally seemed to lead them astray. The faith they preached may have been right, but what happened behind the scenes was not. I doubt Ajora knew not the power of the Lucavi - he seemed quite willing to let Altima take over in order to maintain his resurrected state - else Altima was speaking through him the whole time, which I cannot believe from the dialogue in the final battle as Ajora is there, before becoming Altima.
One small thing of note was the disaster that sunk Murond - it could be considered wether that was natural, or wether the Lucavi may have caused it in a hope to establish Ajora's faith more so they could use it. It is not the most pleasant thought, but it is a possibily in my mind.
No matter how its looked at, the Lucavi and Ajora become intertwined in the ending of the game when you must chase after Alma - reaching as far as Ramza's own family [Dycedarg & Alma]. Another family suffered at the hands of the Lucavi as well - the Tingel family was mired in the holy stone dilemma because of Vormav's possession, causing the death of Izlude, and leaving Meliadoul alone in the world. Ramza and Meliadoul suffered greatly at the hands of the Lucavi.