As you alluded to in your question, there is not one answer that will be true for all mutual funds. In fact, I would argue the question is not specific to mutual funds but can be applied to almost anyone who must make an investment decision: a mutual fund manager, hedge fund manager, or an individual investor.
Even though money going into a company 401(k) retirement savings plan is typically automatically allocated to different funds as we have specified, this is generally not the case for other investment accounts. For example, I also have a Roth IRA in which I have some money from each paycheck direct deposited and it's up to me to decide whether to leave that money in cash or to invest it somewhere else. Every time you invest more money into a mutual fund, the fund manager has the same decision to make.
There are two commonly used mutual fund figures that relate to your question: turnover rate, and cash reserves.
Turnover rate measures the percent of a fund's portfolio that changes every year. For example, a turnover rate of 100% indicates that a fund replaces every asset it held at the beginning of the year with something else at the end of the year – funds with turnover rates greater than 100% average a holding period for a given asset of less than one year, and funds with turnover rates less than 100% average a holding period for a given asset of more than one year.
Cash reserves simply measure the amount of money funds choose to keep as cash instead of investing in other assets.
Another important distinction to make is between actively managed funds and passively managed funds. Passively managed funds are often referred to as "index funds" and have as their goal only to match the returns of a given index or some other benchmark. Actively managed funds on the other hand try to beat the market by exploiting so-called market inefficiencies; e.g. buying undervalued assets, selling overvalued assets, "timing" the market, etc.
To answer your question for a specific fund, I would encourage you to look at the fund's prospectus.
I take as one example of a passively managed fund the Vanguard 500 Index Fund (VFINX), a mutual fund that was created to track the S&P 500. In its prospectus, the fund states that, "to track its target index as closely as possible, the Fund attempts to remain fully invested in stocks". Furthermore, the prospectus states that "the fund's daily cash balance may be invested in one or more Vanguard CMT Funds, which are very low-cost money market funds." Therefore, we would expect both this fund's turnover rate and cash reserves to be extremely low. When we look at its portfolio composition, we see this is true – it is currently at a 4.8% turnover rate and holds 0.0% in short term reserves. Therefore, we can assume this fund is regularly purchasing shares (similar to a dollar cost averaging strategy) instead of holding on to cash and purchasing shares together at a specific time.
For actively managed funds, the picture will tend to look a little different. For example, if we look at the Magellan Fund's portfolio composition, we can see it has a turnover rate of 42%, and holds around .95% in cash/short term reserves. In this case, we can safely guess that trading activity may not be as regular as a passively managed fund, as an active manager attempts to time the market. You may find mutual funds that have much higher cash reserves – perhaps 10% or even more.
Granted, it is impossible to know the exact trading strategy of a mutual fund, and for good reason – if we knew for example, that a fund purchases shares every day at 2:30PM in order to realign with the S&P 500, then sellers of S&P components could up the prices at that time to exploit the mutual fund's trade strategy. Large traders are constantly trying to find ways to conceal their actual trading activity in order to avoid these exact problems.
Finally, I feel obligated to note that it is important to keep in mind that trade frequency is linked to transactions costs – in general, the more frequently an investment manager (whether it be you or a mutual fund manager) executes trades, the more that manager will lose in transactions costs.
