* Jobseekers able to carry over unemployment benefit for 6 months

* Freedom of movement of workers at heart of rules, Brexit vote

* No changes to 'export' of child benefits (Updates after announcement of plans)

STRASBOURG, Dec 13 (Reuters) - The European Commission proposed new rules on Tuesday on what social benefits governments must pay to citizens from other EU states - a key for people concerned about immigration and "benefit tourism".

The package of measures, delayed by nearly a year as Brussels hoped to avoid Britain voting to leave the EU, is designed to clarify rights of EU expatriate workers and states' obligations on unemployment benefits and social security.

The new rules will confirm a series of EU court decisions that backed mainly rich northwestern states in rejecting "benefit tourism" by poor Europeans who did not look for, or find, work after arriving.

"To sum up in one word, it is about fairness," said EU Commissioner for employment and social affair Marianne Thyssen.

Under the proposal, jobseekers will be able to take unemployment benefit with them to a different country for six months, rather than three.

Economically inactive people would, however, not be able to claim social security benefits abroad, setting out that freedom of movement of workers is not a full right to reside for everyone.

EU leaders insist Britain must accept the four basic freedoms - movement of workers, goods, services and capital - if it wants to negotiate continued membership of the single market.

Britain's referendum vote in June was partly driven by rejection of free immigration from the rest of the EU.

One element on which former prime minister David Cameron had secured a promise of change before the Brexit vote is absent. States will continue to have to pay full benefits to children of EU expatriates working in their countries, even if those children live in other countries with lower allowances and living costs.

Thyssen said the payment for such exported benefits was less than 1 percent of child benefit costs.

"It's also difficult to argue with someone who is paying full contributions and taxes as a worker that they should not receive the same benefits," she said.

Taking account of Europe's ageing population, the measures will clarify the responsibilities of states in providing long-term social care, not just medical benefits, for people who have worked and paid into social security systems in other states.

Legal protections will be tightened on social security rules for "posted workers" -- people working on contracts outside their home country. Their general conditions are at the heart of a related piece of legislation which is now being argued over by member states. Rich states say it will let workers from poorer countries undercut wages in their own national economies. (Reporting by Alastair Macdonald and Philip Blenkinsop; Editing by Robin Pomeroy)