Swann made his first competitive return since undergoing elbow surgery while Broad showed little sign of the groin complaint that forced him from Trent Bridge on Monday.
While both failed to grab a wicket, England officials will have been happy to see them get through the session just over two weeks from the opening Test of the summer against New Zealand at Lord's.
Swann was forced home from the reverse tour of New Zealand on March 6 before undergoing surgery in the United States when 14 pieces of bone were removed from his elbow. The 32-year-old last week said his recovery had gone "perfectly" and returned with seven overs at the cost of 19 runs.
Broad was forced off on Monday night, after bowling just three overs, but was able to bowl five more on Tuesday.
While the England pair were unable to take a wicket, the hosts still managed to displace three of Durham's top order to leave them 130 for four, in reply to 320.
England captain Alastair Cook's first competitive innings of the summer was holding Essex together against Hampshire.
Cook was unbeaten on 52 at lunch having watched all seven wickets fall at the other end. Essex were therefore in danger of conceding a first-innings lead, at 144 for seven, despite bowling out Hampshire for 197.
Derbyshire opener Chesney Hughes continued to extend his first-class best innings to an unbeaten 240 as Yorkshire's bowlers were again made to toil. Hughes has so far hit 37 fours and three sixes to help guide his side to 416 for seven.
Leicestershire made a poor start on day one of their match against Gloucestershire as they stumbled to lunch at 67 for five, with Benny Howell taking three for 17.