In First Class cricket - typically towards the end of the final day - it is not uncommon for the fielding team's captain to offer a shake of hands to the batsman and declare the game a draw. This normally happens if the captain does not think their team has a realistic chance of winning the game - for example still needing to take 10 wickets, on a dead pitch, with say 5 overs to go.
Is there an earliest time that a Captain can offer the draw? For example, can it only be made inside the last hour as defined by the Umpires, or could it be made at any time?
As a "case study", in the 2nd Test of England's 2022 tour of the West Indies, at the start of the final day, the match is only 15 overs into the 3rd innings (England batting) with England having a 136 run lead. In all likelihood, baring a spectacular batting collapse, the game is heading for a Draw.
England could choose to try and bat out the day, in which case the West Indies captain would be bowling their bowlers into the ground for essentially no reason; they are that far behind in the game that any total is likely to be unrealistic to chase.
On the flip side, England could be aggressive and decide to go hell for leather and declare, and then try to bowl West Indies out. However, given the lack of any real assistance from the pitch, and having just come off from bowling ~188 overs themselves, that may also be a tiring, fruitless, task - especially given the Player Welfare angle the ECB have been following of late.
So at what point could the fielding captain say "enough is enough, lets call it a draw and go down to the beach"?