The trustee-to-trustee transfer will be in cash, so the mutual funds held in the former employer's account will have to be sold. Open-ended mutual funds only trade at the net asset value of the funds at the end of the trading day. Closed-end mutual funds can trade at a discount or premium to the fund's net asset value and can trade like shares of stock during the trading day. It's far more likely you own open-ended funds in your 401(k). You should be able to log in to the online platform of your previous employer's 401(k) plan and request the exchange of your mutual fund shares for money market mutual fund shares. Choose a nice, conservative money market mutual fund such as a U.S. Treasury fund. Request the transfer toward the end of the U.S. trading day, prior to 4 p.m. Transfer requests placed after 4 p.m. Eastern time will trade the following day.
If you want to control when the shares are sold, then you can sell them yourself in the old plan by moving the money from the mutual fund to a money market mutual fund. Then, when you request the trustee-to-trustee transfer, you're getting out of a low-volatility investment and moving the money into your employer's plan. You can make the same type of decision in the new plan, where you invest the money in a money market mutual fund and then change the investment allocation to your desired allocation on a day you feel is opportunistic for the reallocation.
The conversion to cash has to happen before your previous employer can do the trustee-to-trustee transfer, so you're controlling the asset allocation but the employer controls the timing. There's no guarantee you won't miss a few up days in the market, but then again you were looking to protect against downside risk.
I'm not a huge fan of rolling old 401(k) plans over to the new employer's plan, but then again, I'd like to think I'm a pretty knowledgeable investor. You can gain a lot of flexibility in how the money is invested by transferring it into a traditional or Roth individual retirement account rollover and possibly save money in annual account administration fees and expenses, too.
Figure out why you want to move the money to the new employer's plan. Convenience is one reason. The investment alternatives are another. I'm not saying it's the wrong decision. I just want you to think through why you're making this decision to improve the odds it's the right decision for you.