By Eun-Young Jeong in Seoul and Paul Hannon in London

Factories in Asia and Europe continued their return to normality in June as restrictions designed to contain the coronavirus were lifted, but weak demand still weighed on activity, according to surveys of purchasing managers.

Manufacturing sectors returned to growth in a number of countries, including France, the U.K., Malaysia, Vietnam, Australia and Ireland. Elsewhere, the declines in activity recorded since lockdowns began eased substantially, a sign that the global economy is rebounding from a deep contraction.

"We certainly have a recovery," said Patrick Artus, chief economist at French bank Natixis. "In June, the recovery seems to be faster than we expected."

But while businesses reported that the lifting of restrictions had made it possible to bring workers back to the factory floor and get supplies or raw materials, they said weak demand was holding them back. Manufacturers reported particular weakness in overseas orders and said they had fired workers.

"Even with these gains, production and sentiment remain below pre-pandemic peaks, and persistent weak demand combined with ongoing social distancing measures are likely to act as a drag on the recovery," said Chris Williamson, chief business economist at IHS Markit, the data firm that carries out most of the surveys.

China is the global economy's manufacturing powerhouse and was the first major economy to ease its lockdown, having been the first to fall victim to the virus. The June survey of purchasing managers pointed to an acceleration in activity, a path that other countries will hope to follow in coming months.

China's January lockdown led to an interruption of supplies to other Asian factories, in addition to manufacturers in Asia and Europe. Subsequent lockdowns in other leading manufacturers exacerbated that disruption, while interruptions to air and sea transport have made it difficult for businesses to get their goods to customers.

For Kaiterra, an air-quality tracking-device maker whose manufacturing base is in Shenzhen, China, getting finished products to customers' doorsteps has been one of the biggest challenges this month after production finally returned to pre-Covid levels.

"We're spending a lot more of our time on logistics just trying to help people find better [shipping] deals," said Liam Bates, Kaiterra's chief executive.

Demand for the Switzerland-based company's devices have skyrocketed this year as a result of the pandemic, which increased some people's concerns about air quality.

The company has turned more to sea shipments after airfreight prices surged and it faced wait times of more than two weeks to get packages on flights.

Logistical challenges have also increased the cost of obtaining components. Kaiterra receives around 150 components from up to 20 suppliers across six countries.

The supply side has been less of a headache, said Mr. Bates, adding that the boost in sales helped the company to offset the higher shipping costs.

"What's unusual about this pandemic is everyone is in the same boat. We all know that something crazy is going on," said Mr. Bates. "People are super understanding."

But while logistical bottlenecks are now being cleared, other factories are encountering what is potentially a larger problem: weak demand.

"We had no issues at all getting components, but we didn't get orders from our customers," said Park Jong-jin, head of the planning team at YoungjinIND Ltd., a semiconductor-equipment manufacturer.

The Icheon, South Korea-based company had previously halted production when orders dried up in February and March. The company had to temporarily transfer all its manufacturing staff to take on maintenance and repair roles. Manufacturing restarted in June as new and delayed orders began trickling in.

Data releases over recent weeks have pointed to a strong rebound in consumer spending in a number of large economies, including the U.S. But retailers appear to have met that surge in demand from stocks, with little sign of a boost to factory order books.

Germany is Europe's largest manufacturer, and figures released Wednesday by its statistics office showed a 13.9% increase in retail sales between April and May. But the country's survey of purchasing managers at factories recorded a continued fall in new orders during June.

Factories have responded to shrinking order books by laying off workers. Rising unemployment is likely to weaken demand further, threatening a vicious cycle. That suggests governments and central banks are likely to add to their existing stimulus measures to ensure that the recovery proceeds smoothly, and the global economy avoids a double-dip recession.

Write to Eun-Young Jeong at Eun-Young.Jeong@wsj.com and Paul Hannon at paul.hannon@wsj.com