What incentives are there to move IRA money into a 401k?

Is there an incentive for me to move the IRA money into my 401k as an after-tax contribution?

Not really. You may be confusing that with a Roth conversion.

A Roth IRA conversion is a transfer of retirement assets from a Traditional, SEP, or SIMPLE IRA into a Roth IRA, which creates a taxable event. A Roth IRA conversion can be advantageous for individuals with large traditional IRA accounts who expect their future tax bills to stay at the same level or grow at the time they plan to start withdrawing from their tax-advantaged account, as a Roth IRA allows for tax-free withdrawals of qualified distributions.

If you think your tax rate will be higher during retirement, then rolling over your (pre-tax) IRA into a (post-tax) Roth IRA might be a good idea. WARNING: that rollover is going to be considered current income, and you'll cough up a bundle in taxes. So beware, and do it in stages if -- after analysis -- you think it's worthwhile.

Assuming the plan accepts rollovers, and ignoring any pre-tax vs after-tax vs Roth issues:

A 401(k) plan (or 403(b) or other similar plans) may provide access to special share classes of certain investment options (e.g. mutual funds) that have lower fees than what is available to a standard retail customer (e.g. via an IRA or regular brokerage account). This is not always the case, though, and you need to evaluate the specific investment options available in your plan to see if this applies. If you are invested in a fund in your IRA that has a cheaper class available in your 401(k), you could save some money on fees.

The plan may also offer a loan option, with certain limits such as only allowing you to borrow X% of your balance. Ignoring the question of whether such a loan is a good idea, or better/worse than just withdrawing (potentially with an early withdrawal penalty) from your IRA, rolling an IRA into your 401(k) could increase the amount you are able to borrow.