Chris Gayle was bowled for 37, made up of nine boundaries, and Wavell Hinds lbw for 26.
Frost's misplaced pull off medium-pacer Wavell Hinds went straight to Paul Borrington at mid-wicket to make the score 82-2.
But, in chasing Derbyshire's 164-5, in which Wavell Hinds hit an unbeaten 49 and Garry Park 42, Yorkshire folded.
Morris risks losing batsmen Mahela Jayawardene and Wavell Hinds to the Indian Premier League and the Indian Cricket League respectively.
In reply, Bangladesh initially struggled against the Board XI seam attack of Pedro Collins, Jermaine Lawson, Dwight Washington and Wavell Hinds.
"We are obviously very disappointed that Wavell will be unable to join us this season, " said Derbyshire chief executive Tom Sears.
The hosts started their innings in positive fashion, but lost Wavell hinds for 10 when a Bond delivery uprooted his off-stump.
Wavell Hinds (14) was the other unbeaten batsman at close of play.
BBC: SPORT | Cricket | International Teams | West Indies press home advantage
The England and Wales Cricket Board rejected the registrations of Wavell Hinds, Johannes van der Wath, Andrew Hall, Hamish Marshall and Justin Kemp.
And there was an agnosing half-hour while Wavell Hinds and Lara struggled to get into double figures, Hinds edging Jacques Kallis behind shortly afterwards.
West Indies openers Chris Gayle and Wavell Hinds survived five overs before the close, with Gayle reaching 14 not out and Hinds making five.
That threatened to prove costly as the fourth-wicket pair began to frustrate Derbyshire but Rogers's decision to throw the ball to Wavell Hinds was an inspired one.
Then Wavell Hinds was stumped by Adam Gilchrist off Shane Warne shortly before the close of play, leaving Sarwan and nightwatchman Daren Powell to survive through to stumps.
West Indies were 35 for no wicket when play ended with opener Chris Gayle, who hit 154 in the first innings, nine not out and Wavell Hinds on 22.
Derbyshire still needed 120 to avoid the follow-on, and Wavell Hinds hit two sixes in his 40 before holing out to long-on to give the Bears their first bowling point.
Test opener Wavell Hinds was originally ruled out of the match with an upset stomach but replaced off-spinner Omari Banks in the side after tea with the permission of the hosts.
Wicketkeeper Ajay Ratra, who scored his maiden Test century in the drawn fourth test in Antigua, was caught for three fending a short ball from Dillon straight to Wavell Hinds at forward short-leg.
Madsen lost nightwatchman Jake Needham in the first over of the day, but then shared stands of 117 with Wavell Hinds (54) and 78 with Greg Smith (34) for the fifth and sixth wickets before the declaration came.
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