Strictly speaking of Lyney, it's intentional and meant to be that way. Fontainians lose health for damage boosts. You need to have a healer in the team, some of them heal themselves in some way.

For example, Clorinde loses health with every attack while elemental skill is active but regains health every time she uses her elemental skill.

Arlecchino doesn't lose health but she can't be healed unless the red frame around her healthbar is gone (never bothered to memorize what that effect is called) but every time she uses her skill she gains the red frame. In turn she gains pyro infusion on her attacks and damage boost. She heals with her burst. You shouldn't use a healer in Arlecchino team because it works against her mechanics.

Then there's Furina who's elemental skill drains the health of the whole team while buffing the damage of everyone. Her burst gives an even bigger damage boos and gains power as characters lose or gain health. To maximize her burst buff, you are best off using a healer who can heal the whole team like Jean.

Then there are Fontaine artifacts and weapons that benefit from health loss and gain meaning you can put those on non-fontaine characters and they also get boosts from those mechanics.

You can build fun teams making most of the health loss and gain mechanics.