The visitors, replying to their opponents' first-innings total of 229, started day two on 44 for none and by lunch had moved to 180 for one, with Sam Robson (57 not out) and Joe Denly (77no) having each gone past the half-century mark after the removal of Dawid Malan (32).
Finn, who it was confirmed had been omitted by England for the second Investec Ashes Test against Australia at Lord's and replaced by Tim Bresnan, was set to join his county team-mates at the match on Thursday afternoon.
Alex Lees reached his double century for Yorkshire on day two of their encounter with rock-bottom Derbyshire at Chesterfield.
Lees began the day on 171 and by the interval was still at the crease on 221 not out, with his team having advanced from 367 for two to a massive 509 for three.
Elsewhere in the top tier, Nottinghamshire, who began their second innings on Wednesday needing a huge 427 for victory, were on 78 for two at the break on day four of their meeting with champions Warwickshire at Edgbaston.
In Division Two, things continued to go well for Gloucestershire on day two of their home clash with Worcestershire.
The away outfit had been bowled out for 182 on Wednesday and Gloucestershire were on 193 for two at lunch, with Chris Dent their top scorer having been dismissed with 79 runs to his name.
In another match on day two, basement side Leicestershire's first innings at Essex closed on 365.
Centurion Ned Eckersley (147) was one of five batsmen to fall to the bowling of Graham Napier, whose side were on 24 for none.
There were two second division matches on day four, both heading towards draws.
Hampshire were on 92 for one in their second innings at Kent, who had replied to the visitors' 405 for nine declared with 513.
Meanwhile, table-toppers Lancashire ended up with a first-innings total of 567 all out at Old Trafford in response to Glamorgan's 474.
- Sports & Recreation
- Tim Bresnan