Fair Value Measurements - Schedule of Assets and Liabilities Measured at Fair Value on a Recurring Basis (Details) - USD ($) $ in Millions	Dec. 31, 2021	Sep. 30, 2021
Sponsored funds and separate accounts [Member]
Assets [Abstract]
Investments, at fair value	$ 497.2	$ 368.3
Investments related to long-term incentive plans [Member]
Assets [Abstract]
Investments, at fair value	184.2	160.0
Other equity and debt investments [Member]
Assets [Abstract]
Investments, at fair value	50.5	60.0
Fair Value, Recurring [Member]
Assets [Abstract]
Contingent consideration asset	15.5	19.4
Total Assets Measured at Fair Value	747.4	607.7
Liability [Abstract]
Contingent consideration liabilities	56.7	42.4
Fair Value, Recurring [Member] | Level 1 [Member]
Assets [Abstract]
Contingent consideration asset	0.0	0.0
Total Assets Measured at Fair Value	547.1	404.6
Liability [Abstract]
Contingent consideration liabilities	0.0	0.0
Fair Value, Recurring [Member] | Level 2 [Member]
Assets [Abstract]
Contingent consideration asset	0.0	0.0
Total Assets Measured at Fair Value	60.8	31.7
Liability [Abstract]
Contingent consideration liabilities	0.0	0.0
Fair Value, Recurring [Member] | Level 3 [Member]
Assets [Abstract]
Contingent consideration asset	15.5	19.4
Total Assets Measured at Fair Value	33.7	44.0
Liability [Abstract]
Contingent consideration liabilities	56.7	42.4
Fair Value, Recurring [Member] | NAV as a Practical Expedient [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	105.8	127.4
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member]
Assets [Abstract]
Investments, at fair value	497.2	368.3
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 1 [Member]
Assets [Abstract]
Investments, at fair value	360.0	241.3
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 2 [Member]
Assets [Abstract]
Investments, at fair value	46.7	18.4
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | Level 3 [Member]
Assets [Abstract]
Investments, at fair value	18.2	24.6
Fair Value, Recurring [Member] | Sponsored funds and separate accounts [Member] | NAV as a Practical Expedient [Member]
Assets [Abstract]
Investments, at fair value	72.3	84.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member]
Assets [Abstract]
Investments, at fair value	184.2	160.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 1 [Member]
Assets [Abstract]
Investments, at fair value	183.7	160.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 2 [Member]
Assets [Abstract]
Investments, at fair value	0.5	0.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | Level 3 [Member]
Assets [Abstract]
Investments, at fair value	0.0	0.0
Fair Value, Recurring [Member] | Investments related to long-term incentive plans [Member] | NAV as a Practical Expedient [Member]
Assets [Abstract]
Investments, at fair value	0.0	0.0
Fair Value, Recurring [Member] | Other equity and debt investments [Member]
Assets [Abstract]
Investments, at fair value	50.5	60.0
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 1 [Member]
Assets [Abstract]
Investments, at fair value	3.4	3.3
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 2 [Member]
Assets [Abstract]
Investments, at fair value	13.6	13.3
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | Level 3 [Member]
Assets [Abstract]
Investments, at fair value	0.0	0.0
Fair Value, Recurring [Member] | Other equity and debt investments [Member] | NAV as a Practical Expedient [Member]
Assets [Abstract]
Investments, at fair value	$ 33.5	$ 43.4