Does the Lewis acid have to match the halogen in electrophilic halogenation of benzene

$$\ce{FeBr3}$$ catalyzes bromination and $$\ce{AlCl3}$$ chlorination. I'd like to know whether bromination and chlorination can be done in sequence without changing the catalyst? So - will using the $$\ce{FeBr3}$$ for chlorination and $$\ce{AlCl3}$$ for bromination work? Is it less efficient or slower or is there another reason?

$$\ce{FeBr3 + Cl2 <=> Cl+ + [FeBr3Cl]- <=> Br+ + [FeBr2Cl2]- }\\ \ce{AlCl3 + Br2 <=> Br+ + [AlBrCl3]- <=> Cl+ + [AlBr2Cl2]- }$$