Durham were bowled out for 216 after morning rain at Chester-le-Street had reduced their match with Somerset to 39 overs per side.
Ellie Anderson (3-49) had the hosts wobbling on 49-4 before 89 off 94 balls from captain Hollie Armitage gave the Durham bowlers something to work with, despite Chloe Skelton's 3-16.
Somerset opener Rebecca Odgers knocked 60 runs off the target before Fran Wilson added a run-a-ball 68 before falling to Katie Levick (2-33).
But Amanda Jade-Wellington hit an unbeaten 31 from 27 balls to secure a four-wicket win on the Duckworth-Lewis-Stern method with nine balls to spare.
Emma Lamb continued her good early-season form for Lancashire with 43 off 53 balls at Chelmsford, sharing a 106-run opening stand with Eve Jones (57).
But from 133-1, Essex's Kate Coppack (2-39) and Abtaha Maqsood (3-42) restricted the visitors to 230-8.
In the reply, Hannah Jones took 3-39 and Lamb chipped in with 2-39, but Jodi Grewcock's well-paced undefeated 73, following Alice Macleod's 45 from 47 balls, saw the hosts to a maiden win with 16 balls to spare.
Skipper Bryony Smith stroked 14 fours in a 72-ball 78 before England batter Dunkley - who made 92 against Durham on Wednesday - added a six and seven fours in an unbeaten 102 as Surrey piled up 313-7 at Beckenham.
Davina Perrin set the pace for the Warwickshire reply with 69 at quicker than a run-a-ball, and Natasha Wraith maintained the momentum with 54 from 45 deliveries.
Bethan Ellis then completed a superb chase with an unbeaten 63, as Warwickshire got home with three wickets and 3.3 overs in hand.