Delilah's limit breaks do not work like canon Aerith's.
Here's the logic behind it: there's a line by Yuffie about how the Cetra must've had to fight a lot given the variety and relative abundance of materia, despite how long ago the Cetra as a people disappeared. Going with materia being formed from Cetra knowledge, it stands to reason that not all Cetra were healers like Aerith; some were probably more like red or black mages, possibly blue mages (Enemy Skill materia), maybe even summoners (summon materia, obviously). Now Delilah's skills should be the same as Aerith's if the range of magic possible is hereditary. However, there's no canon indication that Ifalna was a healer. If anything, how Ifalna died indicates she was not likely to be a healer; she was not given an immediately fatal injury, there was time enough to get herself and Aerith to the train station, but she couldn't heal whatever had hurt her.
So what's passed down isn't a specific skillset; just the ability to speak with the Planet and unlock its magic. (Basically what canon already told us was passed down.) What each Cetra learns depends on how they "talk" to it; after all, if two different people talk to the same person, even about the same topic, they're likely to have two different conversations. Aerith has probably wanted to be able to protect and heal people since she was a little girl seeing her mother die, thus leading to her healing-based skillset. Delilah has want to protect and heal people, but also protect herself and punish people who hurt her (not to mention has a space virus in her head), leading to a somewhat different skillset. Air and water still figure into it, as they did in Aerith's.
AerithDelilahHealing WindHealing WindSeal EvilEntrapment
modified "seal evil" + water element + slight HP drain
Breath of the Earth Call of Judgment
drains an enemy's HP to heal entire party
(ex. drain of 500 HP, each member healed for 500) Fury Brand Fury Brand Planet ProtectorHeaven's Vengeance
attack on enemy's HP, dependent on how much
damage party has taken in the battle Pulse of Life
Calamity's Wind
air-based; inflicts a variety of negative status effects on enemies Great Gospel Great Flood
flash flood that completely wipes away lesser enemies
What was Ifalna if not a healer? Again, there's no canon evidence, but there is the implication that it took Tseng a long time to find Aerith with Elmyra--perhaps Ifalna escaped so quickly the trail was lost. But if Ifalna was wounded in the escape, there must have been people after them. How do a woman and little girl who've both been captives for seven years outrun and evade trained soldiers? Haste, perhaps; a proficiency at buff spells. Not that this really matters to Delilah.