Dungeons don't respect timezones, and aren't meaningfully any less active at night than during the day. Still, most SWAT and DRT teams, and definitely most espers, tend to work mostly during the day, barring emergencies. And in this pre-confluence lull, emergencies become less and less frequent. So it's the Asian and Oceanian teams plus a handful of night owls in America who send in the first reports. Some espers and other operatives get alarms on their phones. Some notice it live. Some are woken up to take care of the emergencies that start cropping up all over. Sleep schedules are going to get well and truly fucked.
The volcanoes in this dungeon all erupt. The animals in that zoo-themed dungeon all mutate to get more limbs and become poisonous. These clowns grow sharp teeth. That little idyllic beach victims could just walk out of gets very little warning of the incoming tsunami. A portal spawns not twenty feet away from another one that was already there.
There isn't chaos, though. After decades of experience with dungeons, they've got procedures and systems in place. All guilds coordinate staggered all-hands meetings to brief their teams, databases are updated, teleportation costs drop as they're partially subsidised by various governments and NGOs since people will be needed all over the world, sometimes much farther than they've ever been deployed before. Cross-guild and cross-agency teams are the norm, not the exception, and espers all thrown into a melting pot to work with each other across language and country lines, wherever they're needed.
It's started.