I best remember Bud Black the player as a member of the Giants and Indians, although I do have some cards of him now from his days with the Royals which, like his two games for Seattle as a rookie, were before I got into baseball. One team I don't really associate with him is the Blue Jays; probably because he was only a member of the team for a couple weeks.
The middle of September isn't a time one generally associates with trades. As this Bluebird Banter article from 2012 points out, there have only been eight such trades in Blue Jays history, and this was the only one made to bolster a playoff push. While Black was amidst a good season and would be an upgrade in the fifth spot of the rotation for two crucial games, that was not the primary reason they acquired him.
Black, at that point, had an 11-5 record against Toronto and was scheduled to pitch against the Jays a few days later. The Jays had enough rest days to only need two 5th starters after the September 16, so Black was placed first in the bullpen, making one relief appearance, before making those two starts. He was really acquired so that the Jays didn't have to face him and for some insurance in case Jimmy Key, who was nursing a tender hamstring at the time, did not get better.
The Indians, out of the race since pitchers and catchers reported in February, were just happy to get something for the pending free agent. Toronto sent minor league pitcher Mauro Gozzo and two players to be named later. The second player, named five days later, was another pitcher, Steve Cummings. Three days after that, the Blue Jays fulfilled their obligations by sending yet another pitcher, Alex Sanchez.
Ultimately, the trade was of little consequence for either team as Toronto lost the division to Boston despite Black pitching well, and none of the fringe pitchers panned out for Cleveland. Black went 2-1 with a 4.02 ERA in 15.2 innings over three games; five of those runs were in his one loss. As for the trio of arms he was swapped for, only one saw action in the majors after the trade. Gozzo pitched in exactly two games in each of the 1990, '91, and '92 seasons before getting a chance with the Mets, making 33 appearances between 1993 and '94, his final season.
After the season, Black signed a four-year deal with the Giants for $10,000,000 in a move that befuddled some fans and had others demanding a salary cap. That contract has been pointed to as the one that opened the floodgates for players other than stars to obtain some financial security and then some.