Is sentence "They tend to be being mature" correct? Am I right in thinking that "being" is not needed here?
Perhaps grammatically correct, but odd.
There is no need for a "continuous" form "be being". I can not think of a situation where this helps the meaning. Instead just say "he tends to be immature"
We would say "he is being immature" to mean "he isn't always immature, but he is behaving immaturely now". He "tends to be immature" means "he sometimes behaves immmaturely". Those two meanings can't be easily combined. I would understand "He tends to be being immature" as meaning the same as "He tends to be immature", so the extra word is no needed.
You answered your own question - "being" does not need to be there.
"They tend to be
"Being" is the progressive form (present participle) of the verb "to be", so "be being" makes about as much sense as "she flies flying" (ie no sense at all).
You are right. If we omit "being", the sentence will be correct.
Use the pattern "tend + to be + adjective/participle/noun/noun phrase":
Capital markets in Africa tend to be small and largely under-capitalized.
Older persons tend to be active voters.