How is the word 'relevant' different from other adjectives like it?
Some common synonyms of relevant are applicable, apposite, apropos, germane, material, and pertinent. While all these words mean "relating to or bearing upon the matter in hand," relevant implies a traceable, significant, logical connection.
// found material relevant to her case
When is it sensible to use 'applicable' instead of 'relevant'?
While in some cases nearly identical to relevant, applicable suggests the fitness of bringing a general rule or principle to bear upon a particular case.
// the rule is not applicable in this case
When could 'material' be used to replace 'relevant'?
The words material and relevant can be used in similar contexts, but material implies so close a relationship that it cannot be dispensed with without serious alteration of the case.
// facts material to the investigation
Perhaps contemporize would fit.
to give a modern or contemporary character or setting to; update:
The new production of Romeo and Juliet contemporizes it as the love of two modern teenagers in a Chicago high school.
Dictionary.com
Bring into the mainstream This isn't a single word, which the OP asked for, but I offer it anyway.
In the OP's context:
I am talking about bringing into the mainstream of Indonesian culture certain art theories (contemporary art and postmodernism, mainly) which originated in the Western sphere.
The problem I have with my answer is not that is a phrase, but that I don't know if these art theories are being brought into the mainstream, or will remain only minor tributaries, even among the avant-garde of Indonesia.
mainstream, Oxford Dictionaries
The ideas, attitudes, or activities that are shared by most people and regarded as normal or conventional