Having piled up 2-684, with McSweeney making 101 not out, Lehmann's Northants had bowled out their hosts for 178 with Conway taking 5-36, then reduced them to 3-162 following on with Conway dismissing England Test bat Zak Crawley for the second time in the match.
Monday's fourth and final day dawned with Kent 433 behind and they were soon five down, Conway knocking over nightwatchman Jas Singh.
But then rain and bad light wiped out play from mid-morning to late afternoon.
Eventually, with most spectators having gone, play resumed but Conway could not make the breakthrough. Nor, after it became too dark for the quick to bowl, could part-time offie McSweeney, or anyone else.
Ben Compton, who had been stranded on 99 for four hours, went on to make 114no before Northants gave up.
At Old Trafford Lancashire secured a remarkable win over Derbyshire with former England seamer James Anderson rolling back the years.
The 43-year-old produced a devastating spell to ensure the batting efforts of Marcus Harris did not go to waste
Lancashire had lost 4-13 in the morning, including Harris for 66, to be dismissed for 160 and seemed bound for defeat with Derbyshire needing 137.
Derbys suffered a blow when Harris caught Tasmanian Caleb Jewell for five but had progressed to 3-73 when wickets tumbled with Anderson taking 4-7 in his 29-ball second spell. Derbys were ultimately dismissed for 108 with Anderson effecting the run-out for the final wicket.
In Division One Fergus O'Neill's Nottinghamshire dismissed Glamorgan for 285 to win by 192 runs.
However, the Victorian was not involved as he had substituted after suffering a rib injury, a development permitted under a trial brought in this season.
The extent of O'Neill's injury is unknown but unless it is serious he is unlikely to miss any play. Under the regulations he cannot feature for eight days, but Notts do not play again until April 24.
Similarly Jhye Richardson, who was withdrawn from Yorkshire's match with Hampshire through illness on Saturday, will not miss any cricket. The generosity of the regulation is already drawing criticism.
O'Neill had taken one second innings wicket to give him match figures of 5-66. His replacement, allrounder Lyndon James, took two wickets.
Yorkshire, meanwhile, lost by 214 runs at Leeds, Hampshire having been boosted by earlier innings of 76 and 92 by Jake Lehmann.
Sussex successfully chased down 328 to beat Beau Webster's Warwickshire. An unbeaten 120 by English veteran Jack Leaning built on opener Daniel Hughes' 83 as Sussex won by five wickets. Webster bowled four overs for 23 runs without reward.
A run-fest at The Oval, where fans were encouraged to work remotely with desks provided and superfast wifi, ended in a draw. Surrey and Leicestershire scored 1474 runs for the loss of 24 wickets.
