Dundee Mulligan made his first team debut with
Dundee in the final game of the
2018–19 season against
St Mirren, in which he came on as a substitute. Mulligan would feature in the
Scottish League Cup the following season. In January 2020, Mulligan went on loan with
Scottish League Two league leaders
Cove Rangers until the end of the season. Despite the early curtailment of the season due to the
COVID-19 pandemic, Cove were declared League Two champions after a resolution to finish the season from the
SPFL was passed. In October 2020, Mulligan signed a new contract with Dundee keeping him at the club until 2023, and immediately joined
Scottish League One side
Peterhead on a season-long loan. Mulligan suffered an ankle injury early on in his club debut against
Dundee United, and in November would undergo surgery that would keep him out until the following season. In June 2021, Mulligan would once again join Peterhead on a season-long loan. Despite this, Mulligan would represent Dundee B against Peterhead in the
Scottish Challenge Cup. He would score his first senior goal with the
Blue Toon in a 5–0 win against
Dumbarton. During his successful 2nd stint with the club, Peterhead teammate and media personality
Simon Ferry compared his work-rate to that of
Gareth Bale, and described him as "powerful, strong and direct", and as a "Rolls-Royce" footballer. After a successful first half to the season, Mulligan was recalled early by Dundee in January 2022. He would make his first start for the team in February, in an away victory over
Heart of Midlothian. Mulligan would score his first goal for Dundee against his former club Peterhead in the
Scottish Cup. His first league goal would come in a 3–1 victory over
Hibernian. Mulligan would score the winner for Dundee's first league win of the
2022–23 season in an away win over
Raith Rovers. He followed that up a week later with a fine strike to seal a 4–2 victory over
Arbroath. After injuries to teammates, Mulligan would perform well at right-back in the latter half of the season for the
Dark Blues and was a key contributor in Dundee winning the
Scottish Championship at season's end. On 3 June 2023, Dundee announced that Mulligan had signed a new two-year contract, keeping him at the club until the summer of 2025. In October, it was announced Mulligan needed surgery on a
hernia which would keep him out for several weeks. Mulligan would return in December and would come in and out of the team until being injured again with a hamstring tear in March 2024 which would keep him out for a month or so. Mulligan left Dundee under freedom of contract after the 2024–25 season was completed, and the club subsequently agreed a compensation package with
Hibernian.
Hibernian Mulligan signed a four-year contract with
Hibernian in June 2025. On 24 July 2025, Mulligan made his first competitive appearance for the
Hibees in
Denmark away to
FC Midtjylland in the
UEFA Europa League. On 17 August, Mulligan scored his first Hibernian goal in a
Scottish League Cup victory away to
Livingston. ==International==