Well long story short they did continue to fight on for a short time from the Dutch East Indies after the surrender of the Netherlands in 1940, but by 1942 the Japanese had also completely over run the East Indies, including pretty much the whole Dutch colony.

After the surrender of the Netherlands in 1940, the Dutch colony in the East Indies vowed to continue to fight on with the allies. The whole Dutch East Indies had about 85,000 troops of the Royal Netherlands East Indies Army, of whom about 35,000 were regulars and the rest were militia or volunteer troops, plus about 350 planes, a small fleet of ships and planes from the Royal Dutch Navy, and about 8,000 British, Commonwealth and American troops to strengthen the Dutch garrison. However the majority of these troops were either killed or captured by the Japanese during the East Indies Campaign

After the islands were conquered by the Japanese in mid 1942, the few Dutch troops who evaded capture retreated back to Australia and began to regroup and reequip themselves. Some of the Indigenous Dutch troops who fell back with their units were interned by Australian authorities on suspicion of being pro-Japanese.

After regrouping, small contingents of Dutch East Indies troops fought in the Borneo and New Guinea campaigns alongside Australian and American units and small numbers of Dutch East Indies Air Force planes and pilots were integrated into the RAAF to continue to fight the Japanese. Between 1944 to 1945 MacAuthor also wanted to liberate Java which would have also presumably used Dutch troops, but was ordered not to and the entire Indonesian archipelago was under Japanese control until the end of the war

The largest action they were involved in was the Battle of the Java Sea.

Through January and February 1942 the short lived ABDACOM (Australian British Dutch American Command) had worked to attempt to stall the drive into the Dutch East Indies and towards Singapore, Java, Borneo, and New Guinea.

However superior Japanese mobility, and air support made it an impossible task.

Commanding the ABDA naval forces was Admiral Karl Doorman. And in late Feb when an IJN landing force moved towards Java and Doorman was essentially sent out as a desperate attempt to ambush the landings and slow the advance so what forces remained could evacuate further South, so far the rate of Japanese advance had meant they captured many of the existing Oil wells and rubber plantations almost intact and scooped up many irreplaceable soldiers and civilians too.

The fleet though was at a huge disadvantaged, consisting of ships from 4 different navies and with varying levels of modern equipment such as ship to ship radios, no radar, and no friendly air cover. While the IJN were masters at night surface actions which is what it became after the fleets met in late afternoon.

Doorman was killed when his flagship went down, and while some of the ships got away most of them were hunted down or sacrificed themselves in desperate attacks over the next week.