'assert their right' is correct, grammatically speaking, and it can be used in written English.
For example, "The workers asserted their right to be represented by a union."
What happens when outsiders want to assert their right to that so-called commons?
Nonaligned states assert their right to pursue independent policies away from a commitment to any of the existing alliances headed by the superpowers.
Although Carolingian fortunes waned later in the 9th century, the Carolingians continued to assert their right to protect the church and papacy.
Mr. Coca and Mr. Sadiq assert their right to involve themselves in such issues; their successors have yet to be named.
The Indian asserts his people's troubled past and his right to drink; the doctors assert their right to set social priorities.
Those who have amassed significant wealth or advantage now aggressively assert their right to enjoy it unencumbered by any interference in the form of taxation or redistributive policy.
"The onus will be on rights holders to monitor and assert their rights," he said.
It enabled individuals to publicly assert their rights as equal citizens under the law.
They wanted to assert their rights to their land, prevent deforestation, and continue living ecologically.
They think now is the time to correct social problems, to assert their rights.
Undocumented workers are easy prey for exploitation and unable to assert their rights.
