Are support attached to a character or rolled separately?

Are support attached to a character or rolled separately?

Separately.

Page 7 of the RRG

Support cards represent various vehicles, connections, and logistical aid. When a player plays a support card, they place it faceup in their play area, next to or behind their characters. Supports have repeatable or ongoing effects and stay in play unless an effect or ability discards them.

• Support cards cannot take damage.

• If a support has a die, that die is rolled when the support is activated.

• There is no limit to the number of supports a player can have.

Yes as was mentioned they are rolled separately and therefore are not attached to any characters. They also cannot be damaged and will remain on the board until the game ends or unless the get removed by another effect

Also you can not pay for, or reduce the cost of a support you put into play by switching it with another support like you can with upgrades.