Is there any connection between brand and kalista?
Both being these spirits of vengeance? Kalista of course is the spear of vengeance (or lady of vengeance) and brand being the flame of vengeance.
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According to Brand's lore the fiery vengeance is a being possessing a pirate known as Kegan Rodhe. He is not undead. The being known as brand was a bodiless entity trapped in Ice much like Gnar.
With the Rune Wars being removed from the lore, the most likely guess would be a direct correlation to the Iceborn & Watchers conflict of which 
was a participant.
Brand has "vengeance" attached to his title out of resentment for being locked away, both in ice and in a body. A fire yearns to burn and spread, Brand was contained for quite some time. Consequently, he really wants to set the world on fire, as his selection quote implies.
Well that would Imply that Riot cares about lore.
So I guess the answer is no.
Brand is a creature that is unlike someone such as kalista. Kalista is an undead woman who has been betrayed and in having the barrier of life and death be separated from the shadow isles, she came back as a "spirit of vengeance" as she calls herself. Brand is a being much more. Brand is a living embodiment of destruction as it seems from his lore. He is literally a force of nature in this world. He is not another normal being, but a true part of this world.
There isn't any known connection between the 2. Kalista is an undead spirit,if you were betrayed you can summon her and sell your soul to her,which will cause her to hunt the betrayer. Your soul then becomes a part of her.
Brand on the other hand seems to be an elemental,he was originally a massless fire spirit,who posesses the body of Kegan,brand merely wants to see the world burn and seems to view it as some kind of purification for the earth.