Fair Value Measurements - Schedule of Assets and Liabilities Measured at Fair Value on a Recurring Basis (Details) - USD ($) $ in Millions	Mar. 31, 2024	Sep. 30, 2023
Sponsored funds and separate accounts [Member]
Assets [Abstract]
Investments, at fair value	$ 588.6	$ 630.5
Investments related to long-term incentive plans [Member]
Assets [Abstract]
Investments, at fair value	258.5	191.6
Other equity and debt investments [Member]
Assets [Abstract]
Investments, at fair value	52.6	50.7
Fair Value, Recurring [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	899.7	872.8
Liability [Abstract]
Securities sold short	199.3	158.3
Contingent consideration liabilities	51.4	55.0
Total Liabilities Measured at Fair Value	250.7	213.3
Fair Value, Recurring [Member] \| Level 1 [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	554.7	528.1
Liability [Abstract]
Securities sold short	199.3	158.3
Contingent consideration liabilities	0.0	0.0
Total Liabilities Measured at Fair Value	199.3	158.3
Fair Value, Recurring [Member] \| Level 2 [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	220.6	223.2
Liability [Abstract]
Securities sold short	0.0	0.0
Contingent consideration liabilities	0.0	0.0
Total Liabilities Measured at Fair Value	0.0	0.0
Fair Value, Recurring [Member] \| Level 3 [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	17.4	21.8
Liability [Abstract]
Securities sold short	0.0	0.0
Contingent consideration liabilities	51.4	55.0
Total Liabilities Measured at Fair Value	51.4	55.0
Fair Value, Recurring [Member] \| NAV as a Practical Expedient [Member]
Assets [Abstract]
Total Assets Measured at Fair Value	107.0	99.7
Liability [Abstract]
Securities sold short	0.0	0.0
Contingent consideration liabilities	0.0	0.0
Total Liabilities Measured at Fair Value	0.0	0.0
Fair Value, Recurring [Member] \| Sponsored funds and separate accounts [Member]
Assets [Abstract]
Investments, at fair value	588.6	630.5
Fair Value, Recurring [Member] \| Sponsored funds and separate accounts [Member] \| Level 1 [Member]
Assets [Abstract]
Investments, at fair value	318.3	356.5
Fair Value, Recurring [Member] \| Sponsored funds and separate accounts [Member] \| Level 2 [Member]
Assets [Abstract]
Investments, at fair value	211.6	211.9
Fair Value, Recurring [Member] \| Sponsored funds and separate accounts [Member] \| Level 3 [Member]
Assets [Abstract]
Investments, at fair value	16.4	18.5
Fair Value, Recurring [Member] \| Sponsored funds and separate accounts [Member] \| NAV as a Practical Expedient [Member]
Assets [Abstract]
Investments, at fair value	42.3	43.6
Fair Value, Recurring [Member] \| Investments related to long-term incentive plans [Member]
Assets [Abstract]
Investments, at fair value	258.5	191.6
Fair Value, Recurring [Member] \| Investments related to long-term incentive plans [Member] \| Level 1 [Member]
Assets [Abstract]
Investments, at fair value	232.8	168.2
Fair Value, Recurring [Member] \| Investments related to long-term incentive plans [Member] \| Level 2 [Member]
Assets [Abstract]
Investments, at fair value	0.0	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	25.7	23.4
Fair Value, Recurring [Member] \| Other equity and debt investments [Member]
Assets [Abstract]
Investments, at fair value	52.6	50.7
Fair Value, Recurring [Member] \| Other equity and debt investments [Member] \| Level 1 [Member]
Assets [Abstract]
Investments, at fair value	3.6	3.4
Fair Value, Recurring [Member] \| Other equity and debt investments [Member] \| Level 2 [Member]
Assets [Abstract]
Investments, at fair value	9.0	11.3
Fair Value, Recurring [Member] \| Other equity and debt investments [Member] \| Level 3 [Member]
Assets [Abstract]
Investments, at fair value	1.0	3.3
Fair Value, Recurring [Member] \| Other equity and debt investments [Member] \| NAV as a Practical Expedient [Member]
Assets [Abstract]
Investments, at fair value	$ 39.0	$ 32.7

Fair value portion of probable future economic benefits obtained or controlled by an entity as a result of past transactions or events.

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Amount of liability recognized arising from contingent consideration in a business combination.

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Fair value of obligations arising from derivatives and other contracts sold short.

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Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
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Fair value of financial and nonfinancial obligations.

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Amount of investment in debt security measured at fair value with change in fair value recognized in net income (trading) and investment in equity security measured at fair value with change in fair value recognized in net income (FV-NI).

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Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 210
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