The effect of "Dragon Mastery" which equips a monster on your side of the field with a Dragon-Type "Dragunity" monster activates and starts a Chain Link by targeting the one monster on your side of the field that will be equipped.[1]
When resolving the effect of "Dragon Mastery", if the targeted monster is not face-up on the field, then you do not apply the effect which equips a Dragon-Type "Dragunity" monster to that monster. The Dragon-Type "Dragunity" monster in your hand remains in the hand.[2]
You can use "Dragon Mastery" to equip a "Dragunity" monster to another monster even if that other monster does not have an effect that equips a monster to it, such as "Dragunity Tribus".[4]
References
↑Konami FAQ: Does the effect of "Dragon Mastery" which equips a "Dragunity" card start a Chain Link?
↑Konami FAQ: If the selected monster is not on the field when resolving "Dragon Mastery", then what happens to the Dragon-Type monster in the hand?
↑Konami FAQ: If a monster like "Relinquished" is already equipped with a monster, then can you equip another monster to it with the effect of "Dragon Mastery"?
↑Konami FAQ: With the effect of "Dragon Mastery", can you equip a Dragon-Type monster to a monster that does not have an effect that equips?